Precalculus Examples

$2 x^{4} - 27 x^{3} + 115 x^{2} - 177 x + 63 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $2 x^{4} - 27 x^{3} + 115 x^{2} - 177 x + 63$ using the rational roots test.

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Step 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.

$p = \pm 1, \pm 63, \pm 3, \pm 21, \pm 7, \pm 9$

$q = \pm 1, \pm 2$

Step 1.1.2

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

$\pm 1, \pm 0.5, \pm 63, \pm 31.5, \pm 3, \pm 1.5, \pm 21, \pm 10.5, \pm 7, \pm 3.5, \pm 9, \pm 4.5$

Step 1.1.3

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 1.1.3.1

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{4} - 27 \cdot {0.5}^{3} + 115 \cdot {0.5}^{2} - 177 \cdot 0.5 + 63$

Step 1.1.3.2

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

$2 \cdot 0.0625 - 27 \cdot {0.5}^{3} + 115 \cdot {0.5}^{2} - 177 \cdot 0.5 + 63$

Step 1.1.3.3

Multiply $2$ by $0.0625$ .

$0.125 - 27 \cdot {0.5}^{3} + 115 \cdot {0.5}^{2} - 177 \cdot 0.5 + 63$

Step 1.1.3.4

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

$0.125 - 27 \cdot 0.125 + 115 \cdot {0.5}^{2} - 177 \cdot 0.5 + 63$

Step 1.1.3.5

Multiply $- 27$ by $0.125$ .

$0.125 - 3.375 + 115 \cdot {0.5}^{2} - 177 \cdot 0.5 + 63$

Step 1.1.3.6

Subtract $3.375$ from $0.125$ .

$- 3.25 + 115 \cdot {0.5}^{2} - 177 \cdot 0.5 + 63$

Step 1.1.3.7

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

$- 3.25 + 115 \cdot 0.25 - 177 \cdot 0.5 + 63$

Step 1.1.3.8

Multiply $115$ by $0.25$ .

$- 3.25 + 28.75 - 177 \cdot 0.5 + 63$

Step 1.1.3.9

Add $- 3.25$ and $28.75$ .

$25.5 - 177 \cdot 0.5 + 63$

Step 1.1.3.10

Multiply $- 177$ by $0.5$ .

$25.5 - 88.5 + 63$

Step 1.1.3.11

Subtract $88.5$ from $25.5$ .

$- 63 + 63$

Step 1.1.3.12

Add $- 63$ and $63$ .

$0$

Step 1.1.4

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.

$\frac{2 x^{4} - 27 x^{3} + 115 x^{2} - 177 x + 63}{2 x - 1}$

Step 1.1.5

Divide $2 x^{4} - 27 x^{3} + 115 x^{2} - 177 x + 63$ by $2 x - 1$ .

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Step 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 $0$ .

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

Step 1.1.5.2

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

Step 1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} - x^{3}$

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

Step 1.1.5.5

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

Step 1.1.5.6

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

Step 1.1.5.7

Divide the highest order term in the dividend $- 26 x^{3}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$13 x^{2}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$13 x^{2}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

Step 1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 26 x^{3} + 13 x^{2}$

$x^{3}$

$13 x^{2}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

Step 1.1.5.10

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

$x^{3}$

$13 x^{2}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

Step 1.1.5.11

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

$x^{3}$

$13 x^{2}$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

Step 1.1.5.12

Divide the highest order term in the dividend $102 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$13 x^{2}$

$51 x$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$13 x^{2}$

$51 x$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

$102 x^{2}$

$51 x$

Step 1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $102 x^{2} - 51 x$

$x^{3}$

$13 x^{2}$

$51 x$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

$102 x^{2}$

$51 x$

Step 1.1.5.15

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

$x^{3}$

$13 x^{2}$

$51 x$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

$102 x^{2}$

$51 x$

$126 x$

Step 1.1.5.16

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

$x^{3}$

$13 x^{2}$

$51 x$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

$102 x^{2}$

$51 x$

$126 x$

$63$

Step 1.1.5.17

Divide the highest order term in the dividend $- 126 x$ by the highest order term in divisor $2 x$ .

$x^{3}$

$13 x^{2}$

$51 x$

$63$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

$102 x^{2}$

$51 x$

$126 x$

$63$

Step 1.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$13 x^{2}$

$51 x$

$63$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

$102 x^{2}$

$51 x$

$126 x$

$63$

$126 x$

$63$

Step 1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 126 x + 63$

$x^{3}$

$13 x^{2}$

$51 x$

$63$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

$102 x^{2}$

$51 x$

$126 x$

$63$

$126 x$

$63$

Step 1.1.5.20

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

$x^{3}$

$13 x^{2}$

$51 x$

$63$

$2 x$

$1$

$2 x^{4}$

$27 x^{3}$

$115 x^{2}$

$177 x$

$63$

$2 x^{4}$

$x^{3}$

$26 x^{3}$

$115 x^{2}$

$26 x^{3}$

$13 x^{2}$

$102 x^{2}$

$177 x$

$102 x^{2}$

$51 x$

$126 x$

$63$

$126 x$

$63$

$0$

Step 1.1.5.21

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

$x^{3} - 13 x^{2} + 51 x - 63$

Step 1.1.6

Write $2 x^{4} - 27 x^{3} + 115 x^{2} - 177 x + 63$ as a set of factors.

$(2 x - 1) (x^{3} - 13 x^{2} + 51 x - 63) = 0$

Step 1.2

Factor $x^{3} - 13 x^{2} + 51 x - 63$ using the rational roots test.

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Step 1.2.1

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.

$p = \pm 1, \pm 63, \pm 3, \pm 21, \pm 7, \pm 9$

$q = \pm 1$

Step 1.2.2

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

$\pm 1, \pm 63, \pm 3, \pm 21, \pm 7, \pm 9$

Step 1.2.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 1.2.3.1

Substitute $3$ into the polynomial.

$3^{3} - 13 \cdot 3^{2} + 51 \cdot 3 - 63$

Step 1.2.3.2

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

$27 - 13 \cdot 3^{2} + 51 \cdot 3 - 63$

Step 1.2.3.3

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

$27 - 13 \cdot 9 + 51 \cdot 3 - 63$

Step 1.2.3.4

Multiply $- 13$ by $9$ .

$27 - 117 + 51 \cdot 3 - 63$

Step 1.2.3.5

Subtract $117$ from $27$ .

$- 90 + 51 \cdot 3 - 63$

Step 1.2.3.6

Multiply $51$ by $3$ .

$- 90 + 153 - 63$

Step 1.2.3.7

Add $- 90$ and $153$ .

$63 - 63$

Step 1.2.3.8

Subtract $63$ from $63$ .

$0$

Step 1.2.4

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

$\frac{x^{3} - 13 x^{2} + 51 x - 63}{x - 3}$

Step 1.2.5

Divide $x^{3} - 13 x^{2} + 51 x - 63$ by $x - 3$ .

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Step 1.2.5.1

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

$x$

$3$

$x^{3}$

$13 x^{2}$

$51 x$

$63$

Step 1.2.5.2

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$

Step 1.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			+	$x^{3}$	-	$3 x^{2}$

Step 1.2.5.4

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$

Step 1.2.5.5

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$

Step 1.2.5.6

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$

Step 1.2.5.7

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

				$x^{2}$	-	$10 x$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$

Step 1.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$
					-	$10 x^{2}$	+	$30 x$

Step 1.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} + 30 x$

				$x^{2}$	-	$10 x$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$
					+	$10 x^{2}$	-	$30 x$

Step 1.2.5.10

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

				$x^{2}$	-	$10 x$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$
					+	$10 x^{2}$	-	$30 x$
							+	$21 x$

Step 1.2.5.11

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

				$x^{2}$	-	$10 x$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$
					+	$10 x^{2}$	-	$30 x$
							+	$21 x$	-	$63$

Step 1.2.5.12

Divide the highest order term in the dividend $21 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$	+	$21$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$
					+	$10 x^{2}$	-	$30 x$
							+	$21 x$	-	$63$

Step 1.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$	+	$21$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$
					+	$10 x^{2}$	-	$30 x$
							+	$21 x$	-	$63$
							+	$21 x$	-	$63$

Step 1.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $21 x - 63$

				$x^{2}$	-	$10 x$	+	$21$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$
					+	$10 x^{2}$	-	$30 x$
							+	$21 x$	-	$63$
							-	$21 x$	+	$63$

Step 1.2.5.15

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

				$x^{2}$	-	$10 x$	+	$21$
$x$	-	$3$		$x^{3}$	-	$13 x^{2}$	+	$51 x$	-	$63$
			-	$x^{3}$	+	$3 x^{2}$
					-	$10 x^{2}$	+	$51 x$
					+	$10 x^{2}$	-	$30 x$
							+	$21 x$	-	$63$
							-	$21 x$	+	$63$
										$0$

Step 1.2.5.16

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

$x^{2} - 10 x + 21$

Step 1.2.6

Write $x^{3} - 13 x^{2} + 51 x - 63$ as a set of factors.

$(2 x - 1) ((x - 3) (x^{2} - 10 x + 21)) = 0$

Step 1.3

Factor $x^{2} - 10 x + 21$ using the AC method.

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Step 1.3.1

Factor $x^{2} - 10 x + 21$ using the AC method.

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Step 1.3.1.1

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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Step 1.3.1.2

Write the factored form using these integers.

$(2 x - 1) ((x - 3) ((x - 7) (x - 3))) = 0$

Step 1.3.2

Remove unnecessary parentheses.

$(2 x - 1) ((x - 3) (x - 7) (x - 3)) = 0$

Step 1.4

Combine exponents.

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Step 1.4.1

Combine exponents.

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Step 1.4.1.1

Raise $x - 3$ to the power of $1$ .

$(2 x - 1) ((x - 3) (x - 3) (x - 7)) = 0$

Step 1.4.1.2

Raise $x - 3$ to the power of $1$ .

$(2 x - 1) ((x - 3) (x - 3) (x - 7)) = 0$

Step 1.4.1.3

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

$(2 x - 1) ({(x - 3)}^{1 + 1} (x - 7)) = 0$

Step 1.4.1.4

Add $1$ and $1$ .

$(2 x - 1) ({(x - 3)}^{2} (x - 7)) = 0$

Step 1.4.2

Remove unnecessary parentheses.

$(2 x - 1) {(x - 3)}^{2} (x - 7) = 0$

Step 2

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

$2 x - 1 = 0$

${(x - 3)}^{2} = 0$

$x - 7 = 0$

Step 3

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

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Step 3.1

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

$2 x - 1 = 0$

Step 3.2

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

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Step 3.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 4

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set ${(x - 3)}^{2}$ equal to $0$ .

${(x - 3)}^{2} = 0$

Step 4.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 4.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 5.2

Add $7$ to both sides of the equation.

$x = 7$

Step 6

The final solution is all the values that make $(2 x - 1) {(x - 3)}^{2} (x - 7) = 0$ true.

$x = \frac{1}{2}, 3, 7$