Precalculus Examples

$4 x^{3} - 11 x^{2} - 54 x + 45 = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Factor $4 x^{3} - 11 x^{2} - 54 x + 45$ using the rational roots test.

Tap for more steps...

Step 1.1.1

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.

$p = \pm 1, \pm 45, \pm 3, \pm 15, \pm 5, \pm 9$

$q = \pm 1, \pm 4, \pm 2$

Step 1.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 45, \pm 11.25, \pm 22.5, \pm 3, \pm 0.75, \pm 1.5, \pm 15, \pm 3.75, \pm 7.5, \pm 5, \pm 1.25, \pm 2.5, \pm 9, \pm 2.25, \pm 4.5$

Step 1.1.3

Substitute $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

Tap for more steps...

Step 1.1.3.1

Substitute $- 3$ into the polynomial.

$4 {(- 3)}^{3} - 11 {(- 3)}^{2} - 54 \cdot - 3 + 45$

Step 1.1.3.2

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

$4 \cdot - 27 - 11 {(- 3)}^{2} - 54 \cdot - 3 + 45$

Step 1.1.3.3

Multiply $4$ by $- 27$ .

$- 108 - 11 {(- 3)}^{2} - 54 \cdot - 3 + 45$

Step 1.1.3.4

Raise $- 3$ to the power of $2$ .

$- 108 - 11 \cdot 9 - 54 \cdot - 3 + 45$

Step 1.1.3.5

Multiply $- 11$ by $9$ .

$- 108 - 99 - 54 \cdot - 3 + 45$

Step 1.1.3.6

Subtract $99$ from $- 108$ .

$- 207 - 54 \cdot - 3 + 45$

Step 1.1.3.7

Multiply $- 54$ by $- 3$ .

$- 207 + 162 + 45$

Step 1.1.3.8

Add $- 207$ and $162$ .

$- 45 + 45$

Step 1.1.3.9

Add $- 45$ and $45$ .

$0$

Step 1.1.4

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 11 x^{2} - 54 x + 45}{x + 3}$

Step 1.1.5

Divide $4 x^{3} - 11 x^{2} - 54 x + 45$ by $x + 3$ .

Tap for more steps...

Step 1.1.5.1

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

$x$

$3$

$4 x^{3}$

$11 x^{2}$

$54 x$

$45$

Step 1.1.5.2

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

					$4 x^{2}$
	$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			+	$4 x^{3}$	+	$12 x^{2}$

Step 1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 12 x^{2}$

				$4 x^{2}$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$

Step 1.1.5.5

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

				$4 x^{2}$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$

Step 1.1.5.6

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

				$4 x^{2}$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$

Step 1.1.5.7

Divide the highest order term in the dividend $- 23 x^{2}$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$23 x$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$23 x$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$
					-	$23 x^{2}$	-	$69 x$

Step 1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 23 x^{2} - 69 x$

				$4 x^{2}$	-	$23 x$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$
					+	$23 x^{2}$	+	$69 x$

Step 1.1.5.10

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

				$4 x^{2}$	-	$23 x$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$
					+	$23 x^{2}$	+	$69 x$
							+	$15 x$

Step 1.1.5.11

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

				$4 x^{2}$	-	$23 x$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$
					+	$23 x^{2}$	+	$69 x$
							+	$15 x$	+	$45$

Step 1.1.5.12

Divide the highest order term in the dividend $15 x$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$23 x$	+	$15$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$
					+	$23 x^{2}$	+	$69 x$
							+	$15 x$	+	$45$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$23 x$	+	$15$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$
					+	$23 x^{2}$	+	$69 x$
							+	$15 x$	+	$45$
							+	$15 x$	+	$45$

Step 1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $15 x + 45$

				$4 x^{2}$	-	$23 x$	+	$15$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$
					+	$23 x^{2}$	+	$69 x$
							+	$15 x$	+	$45$
							-	$15 x$	-	$45$

Step 1.1.5.15

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

				$4 x^{2}$	-	$23 x$	+	$15$
$x$	+	$3$		$4 x^{3}$	-	$11 x^{2}$	-	$54 x$	+	$45$
			-	$4 x^{3}$	-	$12 x^{2}$
					-	$23 x^{2}$	-	$54 x$
					+	$23 x^{2}$	+	$69 x$
							+	$15 x$	+	$45$
							-	$15 x$	-	$45$
										$0$

Step 1.1.5.16

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

$4 x^{2} - 23 x + 15$

Step 1.1.6

Write $4 x^{3} - 11 x^{2} - 54 x + 45$ as a set of factors.

$(x + 3) (4 x^{2} - 23 x + 15) = 0$

Step 1.2

Factor by grouping.

Tap for more steps...

Step 1.2.1

Factor by grouping.

Tap for more steps...

Step 1.2.1.1

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 $a \cdot c = 4 \cdot 15 = 60$ and whose sum is $b = - 23$ .

Tap for more steps...

Step 1.2.1.1.1

Factor $- 23$ out of $- 23 x$ .

$(x + 3) (4 x^{2} - 23 x + 15) = 0$

Step 1.2.1.1.2

Rewrite $- 23$ as $- 3$ plus $- 20$

$(x + 3) (4 x^{2} + (- 3 - 20) x + 15) = 0$

Step 1.2.1.1.3

Apply the distributive property.

$(x + 3) (4 x^{2} - 3 x - 20 x + 15) = 0$

Step 1.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.1.2.1

Group the first two terms and the last two terms.

$(x + 3) ((4 x^{2} - 3 x) - 20 x + 15) = 0$

Step 1.2.1.2.2

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

$(x + 3) (x (4 x - 3) - 5 (4 x - 3)) = 0$

Step 1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 3$ .

$(x + 3) ((4 x - 3) (x - 5)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$(x + 3) (4 x - 3) (x - 5) = 0$

Step 2

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

$x + 3 = 0$

$4 x - 3 = 0$

$x - 5 = 0$

Step 3

Set $x + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4

Set $4 x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 4.2

Solve $4 x - 3 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 4.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

Tap for more steps...

Step 4.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 5

Set $x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 6

The final solution is all the values that make $(x + 3) (4 x - 3) (x - 5) = 0$ true.

$x = - 3, \frac{3}{4}, 5$

Step 7