Algebra Examples

$t (x) = x^{4} - 5 x^{3} + 20 x - 16$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 8, \pm 16$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 4, \pm 8, \pm 16$

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.

${(1)}^{4} - 5 {(1)}^{3} + 20 (1) - 16$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 1$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

$1 - 5 {(1)}^{3} + 20 (1) - 16$

Step 4.1.2

One to any power is one.

$1 - 5 \cdot 1 + 20 (1) - 16$

Step 4.1.3

Multiply $- 5$ by $1$ .

$1 - 5 + 20 (1) - 16$

Step 4.1.4

Multiply $20$ by $1$ .

$1 - 5 + 20 - 16$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $5$ from $1$ .

$- 4 + 20 - 16$

Step 4.2.2

Add $- 4$ and $20$ .

$16 - 16$

Step 4.2.3

Subtract $16$ from $16$ .

$0$

Step 5

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

$\frac{x^{4} - 5 x^{3} + 20 x - 16}{x - 1}$

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.

$1$	$1$	$- 5$	$0$	$20$	$- 16$

Step 6.2

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

$1$	$1$	$- 5$	$0$	$20$	$- 16$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(1)$ and place the result of $(1)$ under the next term in the dividend $(- 5)$ .

$1$	$1$	$- 5$	$0$	$20$	$- 16$
		$1$
	$1$

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.

$1$	$1$	$- 5$	$0$	$20$	$- 16$
		$1$
	$1$	$- 4$

Step 6.5

Multiply the newest entry in the result $(- 4)$ by the divisor $(1)$ and place the result of $(- 4)$ under the next term in the dividend $(0)$ .

$1$	$1$	$- 5$	$0$	$20$	$- 16$
		$1$	$- 4$
	$1$	$- 4$

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.

$1$	$1$	$- 5$	$0$	$20$	$- 16$
		$1$	$- 4$
	$1$	$- 4$	$- 4$

Step 6.7

Multiply the newest entry in the result $(- 4)$ by the divisor $(1)$ and place the result of $(- 4)$ under the next term in the dividend $(20)$ .

$1$	$1$	$- 5$	$0$	$20$	$- 16$
		$1$	$- 4$	$- 4$
	$1$	$- 4$	$- 4$

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.

$1$	$1$	$- 5$	$0$	$20$	$- 16$
		$1$	$- 4$	$- 4$
	$1$	$- 4$	$- 4$	$16$

Step 6.9

Multiply the newest entry in the result $(16)$ by the divisor $(1)$ and place the result of $(16)$ under the next term in the dividend $(- 16)$ .

$1$	$1$	$- 5$	$0$	$20$	$- 16$
		$1$	$- 4$	$- 4$	$16$
	$1$	$- 4$	$- 4$	$16$

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.

$1$	$1$	$- 5$	$0$	$20$	$- 16$
		$1$	$- 4$	$- 4$	$16$
	$1$	$- 4$	$- 4$	$16$	$0$

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.

$1 x^{3} + - 4 x^{2} + (- 4) x + 16$

Step 6.12

Simplify the quotient polynomial.

$x^{3} - 4 x^{2} - 4 x + 16$

Step 7

Factor out the greatest common factor from each group.

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Step 7.1

Group the first two terms and the last two terms.

$(x - 1) (x^{3} - 4 x^{2}) - 4 x + 16$

Step 7.2

Factor out the greatest common factor (GCF) from each group.

$(x - 1) + x^{2} (x - 4) - 4 (x - 4)$

$(x - 1) x^{2} (x - 4) - 4 (x - 4)$

Step 8

Factor the polynomial by factoring out the greatest common factor, $x - 4$ .

$(x - 1) (x - 4) (x^{2} - 4)$

Step 9

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

$(x - 1) (x - 4) (x^{2} - 2^{2})$

Step 10

Factor.

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Step 10.1

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 $b = 2$ .

$(x - 1) + (x - 4) ((x + 2) (x - 2))$

Step 10.2

Remove unnecessary parentheses.

$(x - 1) + (x - 4) (x + 2) (x - 2)$

$(x - 1) (x - 4) (x + 2) (x - 2)$

Step 11

Factor the left side of the equation.

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Step 11.1

Factor $x^{4} - 5 x^{3} + 20 x - 16$ using the rational roots test.

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Step 11.1.1

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.

$p = \pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

$q = \pm 1$

Step 11.1.2

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

$\pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

Step 11.1.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 11.1.3.1

Substitute $1$ into the polynomial.

$1^{4} - 5 \cdot 1^{3} + 20 \cdot 1 - 16$

Step 11.1.3.2

Raise $1$ to the power of $4$ .

$1 - 5 \cdot 1^{3} + 20 \cdot 1 - 16$

Step 11.1.3.3

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

$1 - 5 \cdot 1 + 20 \cdot 1 - 16$

Step 11.1.3.4

Multiply $- 5$ by $1$ .

$1 - 5 + 20 \cdot 1 - 16$

Step 11.1.3.5

Subtract $5$ from $1$ .

$- 4 + 20 \cdot 1 - 16$

Step 11.1.3.6

Multiply $20$ by $1$ .

$- 4 + 20 - 16$

Step 11.1.3.7

Add $- 4$ and $20$ .

$16 - 16$

Step 11.1.3.8

Subtract $16$ from $16$ .

$0$

Step 11.1.4

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

$\frac{x^{4} - 5 x^{3} + 20 x - 16}{x - 1}$

Step 11.1.5

Divide $x^{4} - 5 x^{3} + 20 x - 16$ by $x - 1$ .

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Step 11.1.5.1

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

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

Step 11.1.5.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

Step 11.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

Step 11.1.5.4

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

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

Step 11.1.5.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

Step 11.1.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

Step 11.1.5.7

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

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

Step 11.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 11.1.5.9

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

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 11.1.5.10

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

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

Step 11.1.5.11

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

$x^{3}$

$4 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

Step 11.1.5.12

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

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

Step 11.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

$4 x^{2}$

$4 x$

Step 11.1.5.14

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

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

$4 x^{2}$

$4 x$

Step 11.1.5.15

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

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

$4 x^{2}$

$4 x$

$16 x$

Step 11.1.5.16

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

$x^{3}$

$4 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

$4 x^{2}$

$4 x$

$16 x$

$16$

Step 11.1.5.17

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$4 x$

$16$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

$4 x^{2}$

$4 x$

$16 x$

$16$

Step 11.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$4 x$

$16$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

$4 x^{2}$

$4 x$

$16 x$

$16$

$16 x$

$16$

Step 11.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $16 x - 16$

$x^{3}$

$4 x^{2}$

$4 x$

$16$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

$4 x^{2}$

$4 x$

$16 x$

$16$

$16 x$

$16$

Step 11.1.5.20

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

$x^{3}$

$4 x^{2}$

$4 x$

$16$

$x$

$1$

$x^{4}$

$5 x^{3}$

$0 x^{2}$

$20 x$

$16$

$x^{4}$

$x^{3}$

$4 x^{3}$

$0 x^{2}$

$4 x^{3}$

$4 x^{2}$

$20 x$

$4 x^{2}$

$4 x$

$16 x$

$16$

$16 x$

$16$

$0$

Step 11.1.5.21

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

$x^{3} - 4 x^{2} - 4 x + 16$

Step 11.1.6

Write $x^{4} - 5 x^{3} + 20 x - 16$ as a set of factors.

$(x - 1) (x^{3} - 4 x^{2} - 4 x + 16) = 0$

Step 11.2

Factor out the greatest common factor from each group.

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Step 11.2.1

Group the first two terms and the last two terms.

$(x - 1) ((x^{3} - 4 x^{2}) - 4 x + 16) = 0$

Step 11.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 11.3

Factor the polynomial by factoring out the greatest common factor, $x - 4$ .

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

Step 11.4

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

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

Step 11.5

Factor.

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Step 11.5.1

Factor.

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Step 11.5.1.1

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 $b = 2$ .

$(x - 1) ((x - 4) ((x + 2) (x - 2))) = 0$

Step 11.5.1.2

Remove unnecessary parentheses.

$(x - 1) ((x - 4) (x + 2) (x - 2)) = 0$

Step 11.5.2

Remove unnecessary parentheses.

$(x - 1) (x - 4) (x + 2) (x - 2) = 0$

Step 12

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

$x - 1 = 0$

$x - 4 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 13

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

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Step 13.1

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

$x - 1 = 0$

Step 13.2

Add $1$ to both sides of the equation.

$x = 1$

Step 14

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 14.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 14.2

Add $4$ to both sides of the equation.

$x = 4$

Step 15

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

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Step 15.1

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

$x + 2 = 0$

Step 15.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 16

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

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Step 16.1

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

$x - 2 = 0$

Step 16.2

Add $2$ to both sides of the equation.

$x = 2$

Step 17

The final solution is all the values that make $(x - 1) (x - 4) (x + 2) (x - 2) = 0$ true.

$x = 1, 4, - 2, 2$

Step 18