Precalculus Examples

$x^{4} + x^{3} - 42 x^{2}$

Step 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.

$p = \pm 0$

$q = \pm 1$

Step 2

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

$0$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(0)}^{4} + {(0)}^{3} - 42 {(0)}^{2}$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 0$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raising $0$ to any positive power yields $0$ .

$0 + {(0)}^{3} - 42 {(0)}^{2}$

Step 4.1.2

Raising $0$ to any positive power yields $0$ .

$0 + 0 - 42 {(0)}^{2}$

Step 4.1.3

Raising $0$ to any positive power yields $0$ .

$0 + 0 - 42 \cdot 0$

Step 4.1.4

Multiply $- 42$ by $0$ .

$0 + 0 + 0$

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add $0$ and $0$ .

$0 + 0$

Step 4.2.2

Add $0$ and $0$ .

$0$

Step 5

Since $0$ is a known root, divide the polynomial by $x + 0$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} + x^{3} - 42 x^{2}}{x + 0}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$0$	$1$	$1$	$- 42$	$0$	$0$

Step 6.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$0$	$1$	$1$	$- 42$	$0$	$0$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(1)$ .

$0$	$1$	$1$	$- 42$	$0$	$0$
		$0$
	$1$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$1$	$1$	$- 42$	$0$	$0$
		$0$
	$1$	$1$

Step 6.5

Multiply the newest entry in the result $(1)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 42)$ .

$0$	$1$	$1$	$- 42$	$0$	$0$
		$0$	$0$
	$1$	$1$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$1$	$1$	$- 42$	$0$	$0$
		$0$	$0$
	$1$	$1$	$- 42$

Step 6.7

Multiply the newest entry in the result $(- 42)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$1$	$1$	$- 42$	$0$	$0$
		$0$	$0$	$0$
	$1$	$1$	$- 42$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$1$	$1$	$- 42$	$0$	$0$
		$0$	$0$	$0$
	$1$	$1$	$- 42$	$0$

Step 6.9

Multiply the newest entry in the result $(0)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$1$	$1$	$- 42$	$0$	$0$
		$0$	$0$	$0$	$0$
	$1$	$1$	$- 42$	$0$

Step 6.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$0$	$1$	$1$	$- 42$	$0$	$0$
		$0$	$0$	$0$	$0$
	$1$	$1$	$- 42$	$0$	$0$

Step 6.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 1 x^{2} + (- 42) x + 0$

Step 6.12

Simplify the quotient polynomial.

$x^{3} + x^{2} - 42 x$

Step 7

Factor $x$ out of $x^{3} + x^{2} - 42 x$ .

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Step 7.1

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

$(x + 0) (x \cdot x^{2} + x^{2} - 42 x)$

Step 7.2

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

$(x + 0) (x \cdot x^{2} + x \cdot x - 42 x)$

Step 7.3

Factor $x$ out of $- 42 x$ .

$(x + 0) (x \cdot x^{2} + x \cdot x + x \cdot - 42)$

Step 7.4

Factor $x$ out of $x \cdot x^{2} + x \cdot x$ .

$(x + 0) (x (x^{2} + x) + x \cdot - 42)$

Step 7.5

Factor $x$ out of $x (x^{2} + x) + x \cdot - 42$ .

$(x + 0) (x (x^{2} + x - 42))$

Step 8

Factor.

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Step 8.1

Factor $x^{2} + x - 42$ using the AC method.

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Step 8.1.1

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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Step 8.1.2

Write the factored form using these integers.

$(x + 0) (x ((x - 6) (x + 7)))$

Step 8.2

Remove unnecessary parentheses.

$(x + 0) (x (x - 6) (x + 7))$

Step 9

Factor the left side of the equation.

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Step 9.1

Factor $x^{2}$ out of $x^{4} + x^{3} - 42 x^{2}$ .

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Step 9.1.1

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{2} + x^{3} - 42 x^{2} = 0$

Step 9.1.2

Factor $x^{2}$ out of $x^{3}$ .

$x^{2} x^{2} + x^{2} x - 42 x^{2} = 0$

Step 9.1.3

Factor $x^{2}$ out of $- 42 x^{2}$ .

$x^{2} x^{2} + x^{2} x + x^{2} \cdot - 42 = 0$

Step 9.1.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} x$ .

$x^{2} (x^{2} + x) + x^{2} \cdot - 42 = 0$

Step 9.1.5

Factor $x^{2}$ out of $x^{2} (x^{2} + x) + x^{2} \cdot - 42$ .

$x^{2} (x^{2} + x - 42) = 0$

Step 9.2

Factor.

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Step 9.2.1

Factor $x^{2} + x - 42$ using the AC method.

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Step 9.2.1.1

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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Step 9.2.1.2

Write the factored form using these integers.

$x^{2} ((x - 6) (x + 7)) = 0$

Step 9.2.2

Remove unnecessary parentheses.

$x^{2} (x - 6) (x + 7) = 0$

Step 10

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

$x^{2} = 0$

$x - 6 = 0$

$x + 7 = 0$

Step 11

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

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Step 11.1

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

$x^{2} = 0$

Step 11.2

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

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Step 11.2.1

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

$x = \pm \sqrt{0}$

Step 11.2.2

Simplify $\pm \sqrt{0}$ .

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Step 11.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 11.2.2.2

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

$x = \pm 0$

Step 11.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 12

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 12.2

Add $6$ to both sides of the equation.

$x = 6$

Step 13

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

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Step 13.1

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

$x + 7 = 0$

Step 13.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 14

The final solution is all the values that make $x^{2} (x - 6) (x + 7) = 0$ true.

$x = 0, 6, - 7$

Step 15