Algebra Examples

$f (x) = x^{4} + 3 x^{3} - 8 x^{2} + 5 x - 25$

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.

$p = \pm 1, \pm 25, \pm 5$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 25, \pm 5$

Step 3

Substitute $- 5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 5$ is a root of the polynomial.

Tap for more steps...

Step 3.1

Substitute $- 5$ into the polynomial.

${(- 5)}^{4} + 3 {(- 5)}^{3} - 8 {(- 5)}^{2} + 5 \cdot - 5 - 25$

Step 3.2

Raise $- 5$ to the power of $4$ .

$625 + 3 {(- 5)}^{3} - 8 {(- 5)}^{2} + 5 \cdot - 5 - 25$

Step 3.3

Raise $- 5$ to the power of $3$ .

$625 + 3 \cdot - 125 - 8 {(- 5)}^{2} + 5 \cdot - 5 - 25$

Step 3.4

Multiply $3$ by $- 125$ .

$625 - 375 - 8 {(- 5)}^{2} + 5 \cdot - 5 - 25$

Step 3.5

Subtract $375$ from $625$ .

$250 - 8 {(- 5)}^{2} + 5 \cdot - 5 - 25$

Step 3.6

Raise $- 5$ to the power of $2$ .

$250 - 8 \cdot 25 + 5 \cdot - 5 - 25$

Step 3.7

Multiply $- 8$ by $25$ .

$250 - 200 + 5 \cdot - 5 - 25$

Step 3.8

Subtract $200$ from $250$ .

$50 + 5 \cdot - 5 - 25$

Step 3.9

Multiply $5$ by $- 5$ .

$50 - 25 - 25$

Step 3.10

Subtract $25$ from $50$ .

$25 - 25$

Step 3.11

Subtract $25$ from $25$ .

$0$

Step 4

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} + 3 x^{3} - 8 x^{2} + 5 x - 25}{x + 5}$

Step 5

Divide $x^{4} + 3 x^{3} - 8 x^{2} + 5 x - 25$ by $x + 5$ .

Tap for more steps...

Step 5.1

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

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

Step 5.2

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

$x^{3}$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

Step 5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

Step 5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 5 x^{3}$

$x^{3}$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

Step 5.5

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

$x^{3}$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

Step 5.6

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

$x^{3}$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

Step 5.7

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

$x^{3}$

$2 x^{2}$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

Step 5.8

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$2 x^{2}$
$x$	+	$5$		$x^{4}$	+	$3 x^{3}$	-	$8 x^{2}$	+	$5 x$	-	$25$
			-	$x^{4}$	-	$5 x^{3}$
					-	$2 x^{3}$	-	$8 x^{2}$
					-	$2 x^{3}$	-	$10 x^{2}$

Step 5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{3} - 10 x^{2}$

$x^{3}$

$2 x^{2}$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

Step 5.10

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

$x^{3}$

$2 x^{2}$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

Step 5.11

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

$x^{3}$

$2 x^{2}$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

$5 x$

Step 5.12

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

$5 x$

Step 5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

$5 x$

$2 x^{2}$

$10 x$

Step 5.14

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

$5 x$

$2 x^{2}$

$10 x$

Step 5.15

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

$5 x$

$2 x^{2}$

$10 x$

$5 x$

Step 5.16

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

$5 x$

$2 x^{2}$

$10 x$

$5 x$

$25$

Step 5.17

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

$x^{3}$

$2 x^{2}$

$2 x$

$5$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

$5 x$

$2 x^{2}$

$10 x$

$5 x$

$25$

Step 5.18

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$2 x^{2}$	+	$2 x$	-	$5$
$x$	+	$5$		$x^{4}$	+	$3 x^{3}$	-	$8 x^{2}$	+	$5 x$	-	$25$
			-	$x^{4}$	-	$5 x^{3}$
					-	$2 x^{3}$	-	$8 x^{2}$
					+	$2 x^{3}$	+	$10 x^{2}$
							+	$2 x^{2}$	+	$5 x$
							-	$2 x^{2}$	-	$10 x$
									-	$5 x$	-	$25$
									-	$5 x$	-	$25$

Step 5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x - 25$

$x^{3}$

$2 x^{2}$

$2 x$

$5$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

$5 x$

$2 x^{2}$

$10 x$

$5 x$

$25$

$5 x$

$25$

Step 5.20

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

$x^{3}$

$2 x^{2}$

$2 x$

$5$

$x$

$5$

$x^{4}$

$3 x^{3}$

$8 x^{2}$

$5 x$

$25$

$x^{4}$

$5 x^{3}$

$2 x^{3}$

$8 x^{2}$

$2 x^{3}$

$10 x^{2}$

$2 x^{2}$

$5 x$

$2 x^{2}$

$10 x$

$5 x$

$25$

$5 x$

$25$

$0$

Step 5.21

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

$x^{3} - 2 x^{2} + 2 x - 5$

Step 6

Write $x^{4} + 3 x^{3} - 8 x^{2} + 5 x - 25$ as a set of factors.

$f (x) = (x + 5) (x^{3} - 2 x^{2} + 2 x - 5)$