Precalculus Examples

$256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 81$

Step 1

Set $256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 81$ equal to $0$ .

$256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 81 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 81$ using the rational roots test.

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Step 2.1.1.1

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.

$p = \pm 1, \pm 81, \pm 3, \pm 27, \pm 9$

$q = \pm 1, \pm 256, \pm 2, \pm 128, \pm 4, \pm 64, \pm 8, \pm 32, \pm 16$

Step 2.1.1.2

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

$\pm 1, \pm 0.00390625, \pm 0.5, \pm 0.0078125, \pm 0.25, \pm 0.015625, \pm 0.125, \pm 0.03125, \pm 0.0625, \pm 81, \pm 0.31640625, \pm 40.5, \pm 0.6328125, \pm 20.25, \pm 1.265625, \pm 10.125, \pm 2.53125, \pm 5.0625, \pm 3, \pm 0.01171875, \pm 1.5, \pm 0.0234375, \pm 0.75, \pm 0.046875, \pm 0.375, \pm 0.09375, \pm 0.1875, \pm 27, \pm 0.10546875, \pm 13.5, \pm 0.2109375, \pm 6.75, \pm 0.421875, \pm 3.375, \pm 0.84375, \pm 1.6875, \pm 9, \pm 0.03515625, \pm 4.5, \pm 0.0703125, \pm 2.25, \pm 0.140625, \pm 1.125, \pm 0.28125, \pm 0.5625$

Step 2.1.1.3

Substitute $- 0.75$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.75$ is a root of the polynomial.

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Step 2.1.1.3.1

Substitute $- 0.75$ into the polynomial.

$256 {(- 0.75)}^{4} + 768 {(- 0.75)}^{3} + 864 {(- 0.75)}^{2} + 432 \cdot - 0.75 + 81$

Step 2.1.1.3.2

Raise $- 0.75$ to the power of $4$ .

$256 \cdot 0.31640625 + 768 {(- 0.75)}^{3} + 864 {(- 0.75)}^{2} + 432 \cdot - 0.75 + 81$

Step 2.1.1.3.3

Multiply $256$ by $0.31640625$ .

$81 + 768 {(- 0.75)}^{3} + 864 {(- 0.75)}^{2} + 432 \cdot - 0.75 + 81$

Step 2.1.1.3.4

Raise $- 0.75$ to the power of $3$ .

$81 + 768 \cdot - 0.421875 + 864 {(- 0.75)}^{2} + 432 \cdot - 0.75 + 81$

Step 2.1.1.3.5

Multiply $768$ by $- 0.421875$ .

$81 - 324 + 864 {(- 0.75)}^{2} + 432 \cdot - 0.75 + 81$

Step 2.1.1.3.6

Subtract $324$ from $81$ .

$- 243 + 864 {(- 0.75)}^{2} + 432 \cdot - 0.75 + 81$

Step 2.1.1.3.7

Raise $- 0.75$ to the power of $2$ .

$- 243 + 864 \cdot 0.5625 + 432 \cdot - 0.75 + 81$

Step 2.1.1.3.8

Multiply $864$ by $0.5625$ .

$- 243 + 486 + 432 \cdot - 0.75 + 81$

Step 2.1.1.3.9

Add $- 243$ and $486$ .

$243 + 432 \cdot - 0.75 + 81$

Step 2.1.1.3.10

Multiply $432$ by $- 0.75$ .

$243 - 324 + 81$

Step 2.1.1.3.11

Subtract $324$ from $243$ .

$- 81 + 81$

Step 2.1.1.3.12

Add $- 81$ and $81$ .

$0$

Step 2.1.1.4

Since $- 0.75$ is a known root, divide the polynomial by $4 x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 81}{4 x + 3}$

Step 2.1.1.5

Divide $256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 81$ by $4 x + 3$ .

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Step 2.1.1.5.1

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

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

Step 2.1.1.5.2

Divide the highest order term in the dividend $256 x^{4}$ by the highest order term in divisor $4 x$ .

$64 x^{3}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

$64 x^{3}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

Step 2.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $256 x^{4} + 192 x^{3}$

$64 x^{3}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

Step 2.1.1.5.5

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

$64 x^{3}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

Step 2.1.1.5.6

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

$64 x^{3}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

Step 2.1.1.5.7

Divide the highest order term in the dividend $576 x^{3}$ by the highest order term in divisor $4 x$ .

$64 x^{3}$

$144 x^{2}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

$64 x^{3}$

$144 x^{2}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

Step 2.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $576 x^{3} + 432 x^{2}$

$64 x^{3}$

$144 x^{2}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

Step 2.1.1.5.10

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

$64 x^{3}$

$144 x^{2}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

Step 2.1.1.5.11

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

$64 x^{3}$

$144 x^{2}$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

Step 2.1.1.5.12

Divide the highest order term in the dividend $432 x^{2}$ by the highest order term in divisor $4 x$ .

$64 x^{3}$

$144 x^{2}$

$108 x$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

$64 x^{3}$

$144 x^{2}$

$108 x$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

$432 x^{2}$

$324 x$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $432 x^{2} + 324 x$

$64 x^{3}$

$144 x^{2}$

$108 x$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

$432 x^{2}$

$324 x$

Step 2.1.1.5.15

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

$64 x^{3}$

$144 x^{2}$

$108 x$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

$432 x^{2}$

$324 x$

$108 x$

Step 2.1.1.5.16

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

$64 x^{3}$

$144 x^{2}$

$108 x$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

$432 x^{2}$

$324 x$

$108 x$

$81$

Step 2.1.1.5.17

Divide the highest order term in the dividend $108 x$ by the highest order term in divisor $4 x$ .

$64 x^{3}$

$144 x^{2}$

$108 x$

$27$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

$432 x^{2}$

$324 x$

$108 x$

$81$

Step 2.1.1.5.18

Multiply the new quotient term by the divisor.

$64 x^{3}$

$144 x^{2}$

$108 x$

$27$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

$432 x^{2}$

$324 x$

$108 x$

$81$

$108 x$

$81$

Step 2.1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $108 x + 81$

$64 x^{3}$

$144 x^{2}$

$108 x$

$27$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

$432 x^{2}$

$324 x$

$108 x$

$81$

$108 x$

$81$

Step 2.1.1.5.20

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

$64 x^{3}$

$144 x^{2}$

$108 x$

$27$

$4 x$

$3$

$256 x^{4}$

$768 x^{3}$

$864 x^{2}$

$432 x$

$81$

$256 x^{4}$

$192 x^{3}$

$576 x^{3}$

$864 x^{2}$

$576 x^{3}$

$432 x^{2}$

$432 x$

$432 x^{2}$

$324 x$

$108 x$

$81$

$108 x$

$81$

$0$

Step 2.1.1.5.21

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

$64 x^{3} + 144 x^{2} + 108 x + 27$

Step 2.1.1.6

Write $256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 81$ as a set of factors.

$(4 x + 3) (64 x^{3} + 144 x^{2} + 108 x + 27) = 0$

Step 2.1.2

Regroup terms.

$(4 x + 3) (64 x^{3} + 27 + 144 x^{2} + 108 x) = 0$

Step 2.1.3

Rewrite $64 x^{3}$ as ${(4 x)}^{3}$ .

$(4 x + 3) ({(4 x)}^{3} + 27 + 144 x^{2} + 108 x) = 0$

Step 2.1.4

Rewrite $27$ as $3^{3}$ .

$(4 x + 3) ({(4 x)}^{3} + 3^{3} + 144 x^{2} + 108 x) = 0$

Step 2.1.5

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 4 x$ and $b = 3$ .

$(4 x + 3) ((4 x + 3) ({(4 x)}^{2} - 4 x \cdot 3 + 3^{2}) + 144 x^{2} + 108 x) = 0$

Step 2.1.6

Simplify.

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Step 2.1.6.1

Apply the product rule to $4 x$ .

$(4 x + 3) ((4 x + 3) (4^{2} x^{2} - 4 x \cdot 3 + 3^{2}) + 144 x^{2} + 108 x) = 0$

Step 2.1.6.2

Raise $4$ to the power of $2$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} - 4 x \cdot 3 + 3^{2}) + 144 x^{2} + 108 x) = 0$

Step 2.1.6.3

Multiply $4$ by $- 1$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} - 4 x \cdot 3 + 3^{2}) + 144 x^{2} + 108 x) = 0$

Step 2.1.6.4

Multiply $3$ by $- 4$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} - 12 x + 3^{2}) + 144 x^{2} + 108 x) = 0$

Step 2.1.6.5

Raise $3$ to the power of $2$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} - 12 x + 9) + 144 x^{2} + 108 x) = 0$

Step 2.1.7

Factor $36 x$ out of $144 x^{2} + 108 x$ .

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Step 2.1.7.1

Factor $36 x$ out of $144 x^{2}$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} - 12 x + 9) + 36 x (4 x) + 108 x) = 0$

Step 2.1.7.2

Factor $36 x$ out of $108 x$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} - 12 x + 9) + 36 x (4 x) + 36 x (3)) = 0$

Step 2.1.7.3

Factor $36 x$ out of $36 x (4 x) + 36 x (3)$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} - 12 x + 9) + 36 x (4 x + 3)) = 0$

Step 2.1.8

Factor $4 x + 3$ out of $(4 x + 3) (16 x^{2} - 12 x + 9) + 36 x (4 x + 3)$ .

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Step 2.1.8.1

Factor $4 x + 3$ out of $36 x (4 x + 3)$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} - 12 x + 9) + (4 x + 3) (36 x)) = 0$

Step 2.1.8.2

Factor $4 x + 3$ out of $(4 x + 3) (16 x^{2} - 12 x + 9) + (4 x + 3) (36 x)$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} - 12 x + 9 + 36 x)) = 0$

Step 2.1.9

Add $- 12 x$ and $36 x$ .

$(4 x + 3) ((4 x + 3) (16 x^{2} + 24 x + 9)) = 0$

Step 2.1.10

Factor using the perfect square rule.

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Step 2.1.10.1

Factor using the perfect square rule.

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Step 2.1.10.1.1

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$(4 x + 3) ((4 x + 3) ({(4 x)}^{2} + 24 x + 9)) = 0$

Step 2.1.10.1.2

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

$(4 x + 3) ((4 x + 3) ({(4 x)}^{2} + 24 x + 3^{2})) = 0$

Step 2.1.10.1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$24 x = 2 \cdot (4 x) \cdot 3$

Step 2.1.10.1.4

Rewrite the polynomial.

$(4 x + 3) ((4 x + 3) ({(4 x)}^{2} + 2 \cdot (4 x) \cdot 3 + 3^{2})) = 0$

Step 2.1.10.1.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 4 x$ and $b = 3$ .

$(4 x + 3) ((4 x + 3) {(4 x + 3)}^{2}) = 0$

Step 2.1.10.2

Remove unnecessary parentheses.

$(4 x + 3) (4 x + 3) {(4 x + 3)}^{2} = 0$

Step 2.1.11

Combine like factors.

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Step 2.1.11.1

Raise $4 x + 3$ to the power of $1$ .

$(4 x + 3) (4 x + 3) {(4 x + 3)}^{2} = 0$

Step 2.1.11.2

Raise $4 x + 3$ to the power of $1$ .

$(4 x + 3) (4 x + 3) {(4 x + 3)}^{2} = 0$

Step 2.1.11.3

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

${(4 x + 3)}^{1 + 1} {(4 x + 3)}^{2} = 0$

Step 2.1.11.4

Add $1$ and $1$ .

${(4 x + 3)}^{2} {(4 x + 3)}^{2} = 0$

Step 2.1.11.5

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

${(4 x + 3)}^{2 + 2} = 0$

Step 2.1.11.6

Add $2$ and $2$ .

${(4 x + 3)}^{4} = 0$

Step 2.2

Set the $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Step 2.3.2

Divide each term in $4 x = - 3$ by $4$ and simplify.

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Step 2.3.2.1

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

Step 3