Algebra Examples

$6 x^{3} + 25 x^{2} - 24 x + 5$

Step 1

Factor $6 x^{3} + 25 x^{2} - 24 x + 5$ 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.

$p = \pm 1, \pm 5$

$q = \pm 1, \pm 6, \pm 2, \pm 3$

Step 1.2

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

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}, \pm 5, \pm 0.8\overline{3}, \pm 2.5, \pm 1.\overline{6}$

Step 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 1.3.1

Substitute $0.5$ into the polynomial.

$6 \cdot {0.5}^{3} + 25 \cdot {0.5}^{2} - 24 \cdot 0.5 + 5$

Step 1.3.2

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

$6 \cdot 0.125 + 25 \cdot {0.5}^{2} - 24 \cdot 0.5 + 5$

Step 1.3.3

Multiply $6$ by $0.125$ .

$0.75 + 25 \cdot {0.5}^{2} - 24 \cdot 0.5 + 5$

Step 1.3.4

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

$0.75 + 25 \cdot 0.25 - 24 \cdot 0.5 + 5$

Step 1.3.5

Multiply $25$ by $0.25$ .

$0.75 + 6.25 - 24 \cdot 0.5 + 5$

Step 1.3.6

Add $0.75$ and $6.25$ .

$7 - 24 \cdot 0.5 + 5$

Step 1.3.7

Multiply $- 24$ by $0.5$ .

$7 - 12 + 5$

Step 1.3.8

Subtract $12$ from $7$ .

$- 5 + 5$

Step 1.3.9

Add $- 5$ and $5$ .

$0$

Step 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.

$\frac{6 x^{3} + 25 x^{2} - 24 x + 5}{2 x - 1}$

Step 1.5

Divide $6 x^{3} + 25 x^{2} - 24 x + 5$ by $2 x - 1$ .

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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 $0$ .

$2 x$

$1$

$6 x^{3}$

$25 x^{2}$

$24 x$

$5$

Step 1.5.2

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $2 x$ .

					$3 x^{2}$
	$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			+	$6 x^{3}$	-	$3 x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} - 3 x^{2}$

				$3 x^{2}$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$

Step 1.5.5

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

				$3 x^{2}$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$

Step 1.5.6

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

				$3 x^{2}$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$

Step 1.5.7

Divide the highest order term in the dividend $28 x^{2}$ by the highest order term in divisor $2 x$ .

				$3 x^{2}$	+	$14 x$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$14 x$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$
					+	$28 x^{2}$	-	$14 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $28 x^{2} - 14 x$

				$3 x^{2}$	+	$14 x$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$
					-	$28 x^{2}$	+	$14 x$

Step 1.5.10

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

				$3 x^{2}$	+	$14 x$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$
					-	$28 x^{2}$	+	$14 x$
							-	$10 x$

Step 1.5.11

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

				$3 x^{2}$	+	$14 x$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$
					-	$28 x^{2}$	+	$14 x$
							-	$10 x$	+	$5$

Step 1.5.12

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

				$3 x^{2}$	+	$14 x$	-	$5$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$
					-	$28 x^{2}$	+	$14 x$
							-	$10 x$	+	$5$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$14 x$	-	$5$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$
					-	$28 x^{2}$	+	$14 x$
							-	$10 x$	+	$5$
							-	$10 x$	+	$5$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x + 5$

				$3 x^{2}$	+	$14 x$	-	$5$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$
					-	$28 x^{2}$	+	$14 x$
							-	$10 x$	+	$5$
							+	$10 x$	-	$5$

Step 1.5.15

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

				$3 x^{2}$	+	$14 x$	-	$5$
$2 x$	-	$1$		$6 x^{3}$	+	$25 x^{2}$	-	$24 x$	+	$5$
			-	$6 x^{3}$	+	$3 x^{2}$
					+	$28 x^{2}$	-	$24 x$
					-	$28 x^{2}$	+	$14 x$
							-	$10 x$	+	$5$
							+	$10 x$	-	$5$
										$0$

Step 1.5.16

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

$3 x^{2} + 14 x - 5$

Step 1.6

Write $6 x^{3} + 25 x^{2} - 24 x + 5$ as a set of factors.

$(2 x - 1) (3 x^{2} + 14 x - 5)$

Step 2

Factor by grouping.

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Step 2.1

Factor by grouping.

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Step 2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = 14$ .

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Step 2.1.1.1

Factor $14$ out of $14 x$ .

$(2 x - 1) (3 x^{2} + 14 (x) - 5)$

Step 2.1.1.2

Rewrite $14$ as $- 1$ plus $15$

$(2 x - 1) (3 x^{2} + (- 1 + 15) x - 5)$

Step 2.1.1.3

Apply the distributive property.

$(2 x - 1) (3 x^{2} - 1 x + 15 x - 5)$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$(2 x - 1) ((3 x^{2} - 1 x) + 15 x - 5)$

Step 2.1.2.2

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

$(2 x - 1) (x (3 x - 1) + 5 (3 x - 1))$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(2 x - 1) ((3 x - 1) (x + 5))$

Step 2.2

Remove unnecessary parentheses.

$(2 x - 1) (3 x - 1) (x + 5)$