Algebra Examples

$(x^{4} + 5 x^{2} - 36) (2 x^{2} + 9 x - 5) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify $(x^{4} + 5 x^{2} - 36) (2 x^{2} + 9 x - 5)$ .

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Step 1.1.1

Expand $(x^{4} + 5 x^{2} - 36) (2 x^{2} + 9 x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$x^{4} (2 x^{2}) + x^{4} (9 x) + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2

Simplify terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Rewrite using the commutative property of multiplication.

$2 x^{4} x^{2} + x^{4} (9 x) + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.2

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 1.1.2.1.2.1

Move $x^{2}$ .

$2 (x^{2} x^{4}) + x^{4} (9 x) + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{2 + 4} + x^{4} (9 x) + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.2.3

Add $2$ and $4$ .

$2 x^{6} + x^{4} (9 x) + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.3

Rewrite using the commutative property of multiplication.

$2 x^{6} + 9 x^{4} x + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.4

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 1.1.2.1.4.1

Move $x$ .

$2 x^{6} + 9 (x \cdot x^{4}) + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.4.2

Multiply $x$ by $x^{4}$ .

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Step 1.1.2.1.4.2.1

Raise $x$ to the power of $1$ .

$2 x^{6} + 9 (x^{1} x^{4}) + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{6} + 9 x^{1 + 4} + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.4.3

Add $1$ and $4$ .

$2 x^{6} + 9 x^{5} + x^{4} \cdot - 5 + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.5

Move $- 5$ to the left of $x^{4}$ .

$2 x^{6} + 9 x^{5} - 5 \cdot x^{4} + 5 x^{2} (2 x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.6

Rewrite using the commutative property of multiplication.

$2 x^{6} + 9 x^{5} - 5 x^{4} + 5 \cdot 2 x^{2} x^{2} + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.7

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.1.2.1.7.1

Move $x^{2}$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 5 \cdot 2 (x^{2} x^{2}) + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{6} + 9 x^{5} - 5 x^{4} + 5 \cdot 2 x^{2 + 2} + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.7.3

Add $2$ and $2$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 5 \cdot 2 x^{4} + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.8

Multiply $5$ by $2$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 5 x^{2} (9 x) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.9

Rewrite using the commutative property of multiplication.

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 5 \cdot 9 x^{2} x + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.10

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.2.1.10.1

Move $x$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 5 \cdot 9 (x \cdot x^{2}) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.10.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.2.1.10.2.1

Raise $x$ to the power of $1$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 5 \cdot 9 (x^{1} x^{2}) + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 5 \cdot 9 x^{1 + 2} + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.10.3

Add $1$ and $2$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 5 \cdot 9 x^{3} + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.11

Multiply $5$ by $9$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 45 x^{3} + 5 x^{2} \cdot - 5 - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.12

Multiply $- 5$ by $5$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 45 x^{3} - 25 x^{2} - 36 (2 x^{2}) - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.13

Multiply $2$ by $- 36$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 45 x^{3} - 25 x^{2} - 72 x^{2} - 36 (9 x) - 36 \cdot - 5 = 0$

Step 1.1.2.1.14

Multiply $9$ by $- 36$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 45 x^{3} - 25 x^{2} - 72 x^{2} - 324 x - 36 \cdot - 5 = 0$

Step 1.1.2.1.15

Multiply $- 36$ by $- 5$ .

$2 x^{6} + 9 x^{5} - 5 x^{4} + 10 x^{4} + 45 x^{3} - 25 x^{2} - 72 x^{2} - 324 x + 180 = 0$

Step 1.1.2.2

Simplify by adding terms.

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Step 1.1.2.2.1

Add $- 5 x^{4}$ and $10 x^{4}$ .

$2 x^{6} + 9 x^{5} + 5 x^{4} + 45 x^{3} - 25 x^{2} - 72 x^{2} - 324 x + 180 = 0$

Step 1.1.2.2.2

Subtract $72 x^{2}$ from $- 25 x^{2}$ .

$2 x^{6} + 9 x^{5} + 5 x^{4} + 45 x^{3} - 97 x^{2} - 324 x + 180 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $2 x^{6} + 9 x^{5} + 5 x^{4} + 45 x^{3} - 97 x^{2} - 324 x + 180$ using the rational roots test.

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Step 2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 180, \pm 2, \pm 90, \pm 3, \pm 60, \pm 4, \pm 45, \pm 5, \pm 36, \pm 6, \pm 30, \pm 9, \pm 20, \pm 10, \pm 18, \pm 12, \pm 15$

$q = \pm 1, \pm 2$

Step 2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.5, \pm 180, \pm 90, \pm 2, \pm 45, \pm 3, \pm 1.5, \pm 60, \pm 30, \pm 4, \pm 22.5, \pm 5, \pm 2.5, \pm 36, \pm 18, \pm 6, \pm 15, \pm 9, \pm 4.5, \pm 20, \pm 10, \pm 12, \pm 7.5$

Step 2.1.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 2.1.3.1

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{6} + 9 \cdot {0.5}^{5} + 5 \cdot {0.5}^{4} + 45 \cdot {0.5}^{3} - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.2

Raise $0.5$ to the power of $6$ .

$2 \cdot 0.015625 + 9 \cdot {0.5}^{5} + 5 \cdot {0.5}^{4} + 45 \cdot {0.5}^{3} - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.3

Multiply $2$ by $0.015625$ .

$0.03125 + 9 \cdot {0.5}^{5} + 5 \cdot {0.5}^{4} + 45 \cdot {0.5}^{3} - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.4

Raise $0.5$ to the power of $5$ .

$0.03125 + 9 \cdot 0.03125 + 5 \cdot {0.5}^{4} + 45 \cdot {0.5}^{3} - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.5

Multiply $9$ by $0.03125$ .

$0.03125 + 0.28125 + 5 \cdot {0.5}^{4} + 45 \cdot {0.5}^{3} - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.6

Add $0.03125$ and $0.28125$ .

$0.3125 + 5 \cdot {0.5}^{4} + 45 \cdot {0.5}^{3} - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.7

Raise $0.5$ to the power of $4$ .

$0.3125 + 5 \cdot 0.0625 + 45 \cdot {0.5}^{3} - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.8

Multiply $5$ by $0.0625$ .

$0.3125 + 0.3125 + 45 \cdot {0.5}^{3} - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.9

Add $0.3125$ and $0.3125$ .

$0.625 + 45 \cdot {0.5}^{3} - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.10

Raise $0.5$ to the power of $3$ .

$0.625 + 45 \cdot 0.125 - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.11

Multiply $45$ by $0.125$ .

$0.625 + 5.625 - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.12

Add $0.625$ and $5.625$ .

$6.25 - 97 \cdot {0.5}^{2} - 324 \cdot 0.5 + 180$

Step 2.1.3.13

Raise $0.5$ to the power of $2$ .

$6.25 - 97 \cdot 0.25 - 324 \cdot 0.5 + 180$

Step 2.1.3.14

Multiply $- 97$ by $0.25$ .

$6.25 - 24.25 - 324 \cdot 0.5 + 180$

Step 2.1.3.15

Subtract $24.25$ from $6.25$ .

$- 18 - 324 \cdot 0.5 + 180$

Step 2.1.3.16

Multiply $- 324$ by $0.5$ .

$- 18 - 162 + 180$

Step 2.1.3.17

Subtract $162$ from $- 18$ .

$- 180 + 180$

Step 2.1.3.18

Add $- 180$ and $180$ .

$0$

Step 2.1.4

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{6} + 9 x^{5} + 5 x^{4} + 45 x^{3} - 97 x^{2} - 324 x + 180}{2 x - 1}$

Step 2.1.5

Divide $2 x^{6} + 9 x^{5} + 5 x^{4} + 45 x^{3} - 97 x^{2} - 324 x + 180$ by $2 x - 1$ .

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Step 2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

Step 2.1.5.2

Divide the highest order term in the dividend $2 x^{6}$ by the highest order term in divisor $2 x$ .

$x^{5}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

$x^{5}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

Step 2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{6} - x^{5}$

$x^{5}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

Step 2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

Step 2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

Step 2.1.5.7

Divide the highest order term in the dividend $10 x^{5}$ by the highest order term in divisor $2 x$ .

$x^{5}$

$5 x^{4}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

Step 2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{5} - 5 x^{4}$

$x^{5}$

$5 x^{4}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

Step 2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

Step 2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$5 x^{4}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

Step 2.1.5.12

Divide the highest order term in the dividend $10 x^{4}$ by the highest order term in divisor $2 x$ .

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{4} - 5 x^{3}$

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

Step 2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

Step 2.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

Step 2.1.5.17

Divide the highest order term in the dividend $50 x^{3}$ by the highest order term in divisor $2 x$ .

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

Step 2.1.5.18

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

Step 2.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $50 x^{3} - 25 x^{2}$

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

Step 2.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

Step 2.1.5.21

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

Step 2.1.5.22

Divide the highest order term in the dividend $- 72 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$36 x$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

Step 2.1.5.23

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$36 x$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

$72 x^{2}$

$36 x$

Step 2.1.5.24

The expression needs to be subtracted from the dividend, so change all the signs in $- 72 x^{2} + 36 x$

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$36 x$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

$72 x^{2}$

$36 x$

Step 2.1.5.25

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$36 x$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

$72 x^{2}$

$36 x$

$360 x$

Step 2.1.5.26

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$36 x$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

$72 x^{2}$

$36 x$

$360 x$

$180$

Step 2.1.5.27

Divide the highest order term in the dividend $- 360 x$ by the highest order term in divisor $2 x$ .

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$36 x$

$180$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

$72 x^{2}$

$36 x$

$360 x$

$180$

Step 2.1.5.28

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$36 x$

$180$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

$72 x^{2}$

$36 x$

$360 x$

$180$

$360 x$

$180$

Step 2.1.5.29

The expression needs to be subtracted from the dividend, so change all the signs in $- 360 x + 180$

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$36 x$

$180$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

$72 x^{2}$

$36 x$

$360 x$

$180$

$360 x$

$180$

Step 2.1.5.30

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$5 x^{3}$

$25 x^{2}$

$36 x$

$180$

$2 x$

$1$

$2 x^{6}$

$9 x^{5}$

$5 x^{4}$

$45 x^{3}$

$97 x^{2}$

$324 x$

$180$

$2 x^{6}$

$x^{5}$

$10 x^{5}$

$5 x^{4}$

$10 x^{5}$

$5 x^{4}$

$10 x^{4}$

$45 x^{3}$

$10 x^{4}$

$5 x^{3}$

$50 x^{3}$

$97 x^{2}$

$50 x^{3}$

$25 x^{2}$

$72 x^{2}$

$324 x$

$72 x^{2}$

$36 x$

$360 x$

$180$

$360 x$

$180$

$0$

Step 2.1.5.31

Since the remander is $0$ , the final answer is the quotient.

$x^{5} + 5 x^{4} + 5 x^{3} + 25 x^{2} - 36 x - 180$

Step 2.1.6

Write $2 x^{6} + 9 x^{5} + 5 x^{4} + 45 x^{3} - 97 x^{2} - 324 x + 180$ as a set of factors.

$(2 x - 1) (x^{5} + 5 x^{4} + 5 x^{3} + 25 x^{2} - 36 x - 180) = 0$

Step 2.2

Regroup terms.

$(2 x - 1) (x^{5} + 5 x^{3} - 36 x + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.3

Factor $x$ out of $x^{5} + 5 x^{3} - 36 x$ .

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Step 2.3.1

Factor $x$ out of $x^{5}$ .

$(2 x - 1) (x \cdot x^{4} + 5 x^{3} - 36 x + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.3.2

Factor $x$ out of $5 x^{3}$ .

$(2 x - 1) (x \cdot x^{4} + x (5 x^{2}) - 36 x + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.3.3

Factor $x$ out of $- 36 x$ .

$(2 x - 1) (x \cdot x^{4} + x (5 x^{2}) + x \cdot - 36 + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.3.4

Factor $x$ out of $x \cdot x^{4} + x (5 x^{2})$ .

$(2 x - 1) (x (x^{4} + 5 x^{2}) + x \cdot - 36 + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.3.5

Factor $x$ out of $x (x^{4} + 5 x^{2}) + x \cdot - 36$ .

$(2 x - 1) (x (x^{4} + 5 x^{2} - 36) + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.4

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$(2 x - 1) (x ({(x^{2})}^{2} + 5 x^{2} - 36) + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.5

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$(2 x - 1) (x (u^{2} + 5 u - 36) + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.6

Factor $u^{2} + 5 u - 36$ using the AC method.

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Step 2.6.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 36$ and whose sum is $5$ .

$- 4, 9$

Step 2.6.2

Write the factored form using these integers.

$(2 x - 1) (x ((u - 4) (u + 9)) + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.7

Replace all occurrences of $u$ with $x^{2}$ .

$(2 x - 1) (x ((x^{2} - 4) (x^{2} + 9)) + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.8

Rewrite $4$ as $2^{2}$ .

$(2 x - 1) (x ((x^{2} - 2^{2}) (x^{2} + 9)) + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.9

Factor.

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Step 2.9.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$(2 x - 1) (x ((x + 2) (x - 2) (x^{2} + 9)) + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.9.2

Remove unnecessary parentheses.

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 x^{4} + 25 x^{2} - 180) = 0$

Step 2.10

Factor $5$ out of $5 x^{4} + 25 x^{2} - 180$ .

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Step 2.10.1

Factor $5$ out of $5 x^{4}$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 (x^{4}) + 25 x^{2} - 180) = 0$

Step 2.10.2

Factor $5$ out of $25 x^{2}$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 (x^{4}) + 5 (5 x^{2}) - 180) = 0$

Step 2.10.3

Factor $5$ out of $- 180$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 x^{4} + 5 (5 x^{2}) + 5 \cdot - 36) = 0$

Step 2.10.4

Factor $5$ out of $5 x^{4} + 5 (5 x^{2})$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 (x^{4} + 5 x^{2}) + 5 \cdot - 36) = 0$

Step 2.10.5

Factor $5$ out of $5 (x^{4} + 5 x^{2}) + 5 \cdot - 36$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 (x^{4} + 5 x^{2} - 36)) = 0$

Step 2.11

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 ({(x^{2})}^{2} + 5 x^{2} - 36)) = 0$

Step 2.12

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 (u^{2} + 5 u - 36)) = 0$

Step 2.13

Factor $u^{2} + 5 u - 36$ using the AC method.

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Step 2.13.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 36$ and whose sum is $5$ .

$- 4, 9$

Step 2.13.2

Write the factored form using these integers.

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 ((u - 4) (u + 9))) = 0$

Step 2.14

Replace all occurrences of $u$ with $x^{2}$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 ((x^{2} - 4) (x^{2} + 9))) = 0$

Step 2.15

Rewrite $4$ as $2^{2}$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 ((x^{2} - 2^{2}) (x^{2} + 9))) = 0$

Step 2.16

Factor.

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Step 2.16.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 ((x + 2) (x - 2) (x^{2} + 9))) = 0$

Step 2.16.2

Remove unnecessary parentheses.

$(2 x - 1) (x (x + 2) (x - 2) (x^{2} + 9) + 5 (x + 2) (x - 2) (x^{2} + 9)) = 0$

Step 2.17

Factor $(x + 2) (x - 2) (x^{2} + 9)$ out of $x (x + 2) (x - 2) (x^{2} + 9) + 5 (x + 2) (x - 2) (x^{2} + 9)$ .

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Step 2.17.1

Factor $(x + 2) (x - 2) (x^{2} + 9)$ out of $x (x + 2) (x - 2) (x^{2} + 9) + 5 (x + 2) (x - 2) (x^{2} + 9)$ .

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Step 2.17.1.1

Factor $(x + 2) (x - 2) (x^{2} + 9)$ out of $x (x + 2) (x - 2) (x^{2} + 9)$ .

$(2 x - 1) ((x + 2) (x - 2) (x^{2} + 9) (x) + 5 (x + 2) (x - 2) (x^{2} + 9)) = 0$

Step 2.17.1.2

Factor $(x + 2) (x - 2) (x^{2} + 9)$ out of $5 (x + 2) (x - 2) (x^{2} + 9)$ .

$(2 x - 1) ((x + 2) (x - 2) (x^{2} + 9) (x) + (x + 2) (x - 2) (x^{2} + 9) (5)) = 0$

Step 2.17.1.3

Factor $(x + 2) (x - 2) (x^{2} + 9)$ out of $(x + 2) (x - 2) (x^{2} + 9) (x) + (x + 2) (x - 2) (x^{2} + 9) (5)$ .

$(2 x - 1) ((x + 2) (x - 2) (x^{2} + 9) (x + 5)) = 0$

Step 2.17.2

Remove unnecessary parentheses.

$(2 x - 1) (x + 2) (x - 2) (x^{2} + 9) (x + 5) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 1 = 0$

$x + 2 = 0$

$x - 2 = 0$

$x^{2} + 9 = 0$

$x + 5 = 0$

Step 4

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 4.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 4.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 4.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 7

Set $x^{2} + 9$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Step 7.2

Solve $x^{2} + 9 = 0$ for $x$ .

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Step 7.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 7.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 9}$

Step 7.2.3

Simplify $\pm \sqrt{- 9}$ .

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Step 7.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 7.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 7.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{9}$

Step 7.2.3.4

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

Step 7.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 3$

Step 7.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 7.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3 i$

Step 7.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3 i$

Step 7.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3 i, - 3 i$

Step 8

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 8.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 9

The final solution is all the values that make $(2 x - 1) (x + 2) (x - 2) (x^{2} + 9) (x + 5) = 0$ true.

$x = \frac{1}{2}, - 2, 2, 3 i, - 3 i, - 5$