Precalculus Examples

$7 x^{3} - 43 x^{2} + 69 x - 9$

Step 1

Factor $7 x^{3} - 43 x^{2} + 69 x - 9$ using the rational roots test.

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Step 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.

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

$q = \pm 1, \pm 7$

Step 1.2

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

$\pm 1, \pm 0.\overline{142857}, \pm 9, \pm 1.\overline{285714}, \pm 3, \pm 0.\overline{428571}$

Step 1.3

Substitute $0.\overline{142857}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{142857}$ is a root of the polynomial.

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Step 1.3.1

Substitute $0.\overline{142857}$ into the polynomial.

$7 \cdot {0.\overline{142857}}^{3} - 43 \cdot {0.\overline{142857}}^{2} + 69 \cdot 0.\overline{142857} - 9$

Step 1.3.2

Raise $0.\overline{142857}$ to the power of $3$ .

$7 \cdot 0.00291545 - 43 \cdot {0.\overline{142857}}^{2} + 69 \cdot 0.\overline{142857} - 9$

Step 1.3.3

Multiply $7$ by $0.00291545$ .

$0.02040816 - 43 \cdot {0.\overline{142857}}^{2} + 69 \cdot 0.\overline{142857} - 9$

Step 1.3.4

Raise $0.\overline{142857}$ to the power of $2$ .

$0.02040816 - 43 \cdot 0.02040816 + 69 \cdot 0.\overline{142857} - 9$

Step 1.3.5

Multiply $- 43$ by $0.02040816$ .

$0.02040816 - 0.87755102 + 69 \cdot 0.\overline{142857} - 9$

Step 1.3.6

Subtract $0.87755102$ from $0.02040816$ .

$- 0.85714285 + 69 \cdot 0.\overline{142857} - 9$

Step 1.3.7

Multiply $69$ by $0.\overline{142857}$ .

$- 0.85714285 + 9.\overline{857142} - 9$

Step 1.3.8

Add $- 0.85714285$ and $9.\overline{857142}$ .

$9 - 9$

Step 1.3.9

Subtract $9$ from $9$ .

$0$

Step 1.4

Since $0.\overline{142857}$ is a known root, divide the polynomial by $7 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{7 x^{3} - 43 x^{2} + 69 x - 9}{7 x - 1}$

Step 1.5

Divide $7 x^{3} - 43 x^{2} + 69 x - 9$ by $7 x - 1$ .

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Step 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 $0$ .

$7 x$

$1$

$7 x^{3}$

$43 x^{2}$

$69 x$

$9$

Step 1.5.2

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $7 x$ .

					$x^{2}$
	$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			+	$7 x^{3}$	-	$x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} - x^{2}$

				$x^{2}$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$

Step 1.5.5

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

				$x^{2}$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$

Step 1.5.6

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

				$x^{2}$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$

Step 1.5.7

Divide the highest order term in the dividend $- 42 x^{2}$ by the highest order term in divisor $7 x$ .

				$x^{2}$	-	$6 x$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$
					-	$42 x^{2}$	+	$6 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 42 x^{2} + 6 x$

				$x^{2}$	-	$6 x$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$
					+	$42 x^{2}$	-	$6 x$

Step 1.5.10

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

				$x^{2}$	-	$6 x$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$
					+	$42 x^{2}$	-	$6 x$
							+	$63 x$

Step 1.5.11

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

				$x^{2}$	-	$6 x$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$
					+	$42 x^{2}$	-	$6 x$
							+	$63 x$	-	$9$

Step 1.5.12

Divide the highest order term in the dividend $63 x$ by the highest order term in divisor $7 x$ .

				$x^{2}$	-	$6 x$	+	$9$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$
					+	$42 x^{2}$	-	$6 x$
							+	$63 x$	-	$9$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	+	$9$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$
					+	$42 x^{2}$	-	$6 x$
							+	$63 x$	-	$9$
							+	$63 x$	-	$9$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $63 x - 9$

				$x^{2}$	-	$6 x$	+	$9$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$
					+	$42 x^{2}$	-	$6 x$
							+	$63 x$	-	$9$
							-	$63 x$	+	$9$

Step 1.5.15

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

				$x^{2}$	-	$6 x$	+	$9$
$7 x$	-	$1$		$7 x^{3}$	-	$43 x^{2}$	+	$69 x$	-	$9$
			-	$7 x^{3}$	+	$x^{2}$
					-	$42 x^{2}$	+	$69 x$
					+	$42 x^{2}$	-	$6 x$
							+	$63 x$	-	$9$
							-	$63 x$	+	$9$
										$0$

Step 1.5.16

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

$x^{2} - 6 x + 9$

Step 1.6

Write $7 x^{3} - 43 x^{2} + 69 x - 9$ as a set of factors.

$(7 x - 1) (x^{2} - 6 x + 9)$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$(7 x - 1) (x^{2} - 6 x + 3^{2})$

Step 2.2

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

$6 x = 2 \cdot x \cdot 3$

Step 2.3

Rewrite the polynomial.

$(7 x - 1) (x^{2} - 2 \cdot x \cdot 3 + 3^{2})$

Step 2.4

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

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