Precalculus Examples

$4 x^{5} - 16 x^{4} + 17 x^{3} - 19 x^{2} + 13 x - 3 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$4 x^{5} + 17 x^{3} + 13 x - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.2

Factor $x$ out of $4 x^{5} + 17 x^{3} + 13 x$ .

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Step 1.2.1

Factor $x$ out of $4 x^{5}$ .

$x (4 x^{4}) + 17 x^{3} + 13 x - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.2.2

Factor $x$ out of $17 x^{3}$ .

$x (4 x^{4}) + x (17 x^{2}) + 13 x - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.2.3

Factor $x$ out of $13 x$ .

$x (4 x^{4}) + x (17 x^{2}) + x \cdot 13 - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.2.4

Factor $x$ out of $x (4 x^{4}) + x (17 x^{2})$ .

$x (4 x^{4} + 17 x^{2}) + x \cdot 13 - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.2.5

Factor $x$ out of $x (4 x^{4} + 17 x^{2}) + x \cdot 13$ .

$x (4 x^{4} + 17 x^{2} + 13) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (4 {(x^{2})}^{2} + 17 x^{2} + 13) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (4 u^{2} + 17 u + 13) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.5

Factor by grouping.

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Step 1.5.1

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 $a \cdot c = 4 \cdot 13 = 52$ and whose sum is $b = 17$ .

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Step 1.5.1.1

Factor $17$ out of $17 u$ .

$x (4 u^{2} + 17 (u) + 13) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.5.1.2

Rewrite $17$ as $4$ plus $13$

$x (4 u^{2} + (4 + 13) u + 13) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.5.1.3

Apply the distributive property.

$x (4 u^{2} + 4 u + 13 u + 13) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.5.2

Factor out the greatest common factor from each group.

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Step 1.5.2.1

Group the first two terms and the last two terms.

$x ((4 u^{2} + 4 u) + 13 u + 13) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.5.2.2

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

$x (4 u (u + 1) + 13 (u + 1)) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.5.3

Factor the polynomial by factoring out the greatest common factor, $u + 1$ .

$x ((u + 1) (4 u + 13)) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.6

Factor.

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Step 1.6.1

Replace all occurrences of $u$ with $x^{2}$ .

$x ((x^{2} + 1) (4 x^{2} + 13)) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.6.2

Remove unnecessary parentheses.

$x (x^{2} + 1) (4 x^{2} + 13) - 16 x^{4} - 19 x^{2} - 3 = 0$

Step 1.7

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (x^{2} + 1) (4 x^{2} + 13) - 16 {(x^{2})}^{2} - 19 x^{2} - 3 = 0$

Step 1.8

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (x^{2} + 1) (4 x^{2} + 13) - 16 u^{2} - 19 u - 3 = 0$

Step 1.9

Factor by grouping.

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Step 1.9.1

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 $a \cdot c = - 16 \cdot - 3 = 48$ and whose sum is $b = - 19$ .

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Step 1.9.1.1

Factor $- 19$ out of $- 19 u$ .

$x (x^{2} + 1) (4 x^{2} + 13) - 16 u^{2} - 19 u - 3 = 0$

Step 1.9.1.2

Rewrite $- 19$ as $- 3$ plus $- 16$

$x (x^{2} + 1) (4 x^{2} + 13) - 16 u^{2} + (- 3 - 16) u - 3 = 0$

Step 1.9.1.3

Apply the distributive property.

$x (x^{2} + 1) (4 x^{2} + 13) - 16 u^{2} - 3 u - 16 u - 3 = 0$

Step 1.9.2

Factor out the greatest common factor from each group.

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Step 1.9.2.1

Group the first two terms and the last two terms.

$x (x^{2} + 1) (4 x^{2} + 13) - 16 u^{2} - 3 u - 16 u - 3 = 0$

Step 1.9.2.2

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

$x (x^{2} + 1) (4 x^{2} + 13) + u (- 16 u - 3) + 1 (- 16 u - 3) = 0$

Step 1.9.3

Factor the polynomial by factoring out the greatest common factor, $- 16 u - 3$ .

$x (x^{2} + 1) (4 x^{2} + 13) + (- 16 u - 3) (u + 1) = 0$

Step 1.10

Replace all occurrences of $u$ with $x^{2}$ .

$x (x^{2} + 1) (4 x^{2} + 13) + (- 16 x^{2} - 3) (x^{2} + 1) = 0$

Step 1.11

Factor $x^{2} + 1$ out of $x (x^{2} + 1) (4 x^{2} + 13) + (- 16 x^{2} - 3) (x^{2} + 1)$ .

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Step 1.11.1

Factor $x^{2} + 1$ out of $x (x^{2} + 1) (4 x^{2} + 13)$ .

$(x^{2} + 1) (x (4 x^{2} + 13)) + (- 16 x^{2} - 3) (x^{2} + 1) = 0$

Step 1.11.2

Factor $x^{2} + 1$ out of $(- 16 x^{2} - 3) (x^{2} + 1)$ .

$(x^{2} + 1) (x (4 x^{2} + 13)) + (x^{2} + 1) (- 16 x^{2} - 3) = 0$

Step 1.11.3

Factor $x^{2} + 1$ out of $(x^{2} + 1) (x (4 x^{2} + 13)) + (x^{2} + 1) (- 16 x^{2} - 3)$ .

$(x^{2} + 1) (x (4 x^{2} + 13) - 16 x^{2} - 3) = 0$

Step 1.12

Apply the distributive property.

$(x^{2} + 1) (x (4 x^{2}) + x \cdot 13 - 16 x^{2} - 3) = 0$

Step 1.13

Rewrite using the commutative property of multiplication.

$(x^{2} + 1) (4 x \cdot x^{2} + x \cdot 13 - 16 x^{2} - 3) = 0$

Step 1.14

Move $13$ to the left of $x$ .

$(x^{2} + 1) (4 x \cdot x^{2} + 13 \cdot x - 16 x^{2} - 3) = 0$

Step 1.15

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.15.1

Move $x^{2}$ .

$(x^{2} + 1) (4 (x^{2} x) + 13 \cdot x - 16 x^{2} - 3) = 0$

Step 1.15.2

Multiply $x^{2}$ by $x$ .

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Step 1.15.2.1

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

$(x^{2} + 1) (4 (x^{2} x) + 13 \cdot x - 16 x^{2} - 3) = 0$

Step 1.15.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{2} + 1) (4 x^{2 + 1} + 13 \cdot x - 16 x^{2} - 3) = 0$

Step 1.15.3

Add $2$ and $1$ .

$(x^{2} + 1) (4 x^{3} + 13 \cdot x - 16 x^{2} - 3) = 0$

$(x^{2} + 1) (4 x^{3} + 13 x - 16 x^{2} - 3) = 0$

Step 1.16

Reorder terms.

$(x^{2} + 1) (4 x^{3} - 16 x^{2} + 13 x - 3) = 0$

Step 1.17

Factor.

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Step 1.17.1

Rewrite $4 x^{3} - 16 x^{2} + 13 x - 3$ in a factored form.

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Step 1.17.1.1

Factor $4 x^{3} - 16 x^{2} + 13 x - 3$ using the rational roots test.

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Step 1.17.1.1.1

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.

$p = \pm 1, \pm 3$

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

Step 1.17.1.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 3, \pm 0.75, \pm 1.5$

Step 1.17.1.1.3

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 1.17.1.1.3.1

Substitute $0.5$ into the polynomial.

$4 \cdot {0.5}^{3} - 16 \cdot {0.5}^{2} + 13 \cdot 0.5 - 3$

Step 1.17.1.1.3.2

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

$4 \cdot 0.125 - 16 \cdot {0.5}^{2} + 13 \cdot 0.5 - 3$

Step 1.17.1.1.3.3

Multiply $4$ by $0.125$ .

$0.5 - 16 \cdot {0.5}^{2} + 13 \cdot 0.5 - 3$

Step 1.17.1.1.3.4

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

$0.5 - 16 \cdot 0.25 + 13 \cdot 0.5 - 3$

Step 1.17.1.1.3.5

Multiply $- 16$ by $0.25$ .

$0.5 - 4 + 13 \cdot 0.5 - 3$

Step 1.17.1.1.3.6

Subtract $4$ from $0.5$ .

$- 3.5 + 13 \cdot 0.5 - 3$

Step 1.17.1.1.3.7

Multiply $13$ by $0.5$ .

$- 3.5 + 6.5 - 3$

Step 1.17.1.1.3.8

Add $- 3.5$ and $6.5$ .

$3 - 3$

Step 1.17.1.1.3.9

Subtract $3$ from $3$ .

$0$

Step 1.17.1.1.4

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.

$\frac{4 x^{3} - 16 x^{2} + 13 x - 3}{2 x - 1}$

Step 1.17.1.1.5

Divide $4 x^{3} - 16 x^{2} + 13 x - 3$ by $2 x - 1$ .

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Step 1.17.1.1.5.1

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

$2 x$

$1$

$4 x^{3}$

$16 x^{2}$

$13 x$

$3$

Step 1.17.1.1.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$

Step 1.17.1.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			+	$4 x^{3}$	-	$2 x^{2}$

Step 1.17.1.1.5.4

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$

Step 1.17.1.1.5.5

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$

Step 1.17.1.1.5.6

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$

Step 1.17.1.1.5.7

Divide the highest order term in the dividend $- 14 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$7 x$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$

Step 1.17.1.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$7 x$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$
					-	$14 x^{2}$	+	$7 x$

Step 1.17.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x^{2} + 7 x$

				$2 x^{2}$	-	$7 x$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$
					+	$14 x^{2}$	-	$7 x$

Step 1.17.1.1.5.10

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

				$2 x^{2}$	-	$7 x$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$
					+	$14 x^{2}$	-	$7 x$
							+	$6 x$

Step 1.17.1.1.5.11

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

				$2 x^{2}$	-	$7 x$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$
					+	$14 x^{2}$	-	$7 x$
							+	$6 x$	-	$3$

Step 1.17.1.1.5.12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$7 x$	+	$3$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$
					+	$14 x^{2}$	-	$7 x$
							+	$6 x$	-	$3$

Step 1.17.1.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$7 x$	+	$3$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$
					+	$14 x^{2}$	-	$7 x$
							+	$6 x$	-	$3$
							+	$6 x$	-	$3$

Step 1.17.1.1.5.14

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

				$2 x^{2}$	-	$7 x$	+	$3$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$
					+	$14 x^{2}$	-	$7 x$
							+	$6 x$	-	$3$
							-	$6 x$	+	$3$

Step 1.17.1.1.5.15

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

				$2 x^{2}$	-	$7 x$	+	$3$
$2 x$	-	$1$		$4 x^{3}$	-	$16 x^{2}$	+	$13 x$	-	$3$
			-	$4 x^{3}$	+	$2 x^{2}$
					-	$14 x^{2}$	+	$13 x$
					+	$14 x^{2}$	-	$7 x$
							+	$6 x$	-	$3$
							-	$6 x$	+	$3$
										$0$

Step 1.17.1.1.5.16

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

$2 x^{2} - 7 x + 3$

Step 1.17.1.1.6

Write $4 x^{3} - 16 x^{2} + 13 x - 3$ as a set of factors.

$(x^{2} + 1) ((2 x - 1) (2 x^{2} - 7 x + 3)) = 0$

Step 1.17.1.2

Factor by grouping.

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Step 1.17.1.2.1

Factor by grouping.

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Step 1.17.1.2.1.1

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

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Step 1.17.1.2.1.1.1

Factor $- 7$ out of $- 7 x$ .

$(x^{2} + 1) ((2 x - 1) (2 x^{2} - 7 x + 3)) = 0$

Step 1.17.1.2.1.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$(x^{2} + 1) ((2 x - 1) (2 x^{2} + (- 1 - 6) x + 3)) = 0$

Step 1.17.1.2.1.1.3

Apply the distributive property.

$(x^{2} + 1) ((2 x - 1) (2 x^{2} - 1 x - 6 x + 3)) = 0$

Step 1.17.1.2.1.2

Factor out the greatest common factor from each group.

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Step 1.17.1.2.1.2.1

Group the first two terms and the last two terms.

$(x^{2} + 1) ((2 x - 1) ((2 x^{2} - 1 x) - 6 x + 3)) = 0$

Step 1.17.1.2.1.2.2

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

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

Step 1.17.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Step 1.17.1.2.2

Remove unnecessary parentheses.

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

Step 1.17.1.3

Combine like factors.

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Step 1.17.1.3.1

Raise $2 x - 1$ to the power of $1$ .

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

Step 1.17.1.3.2

Raise $2 x - 1$ to the power of $1$ .

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

Step 1.17.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.17.1.3.4

Add $1$ and $1$ .

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

Step 1.17.2

Remove unnecessary parentheses.

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

Step 2

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

$x^{2} + 1 = 0$

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

$x - 3 = 0$

Step 3

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 3.2

Solve $x^{2} + 1 = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 3.2.2

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

$x = \pm \sqrt{- 1}$

Step 3.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 4

Set ${(2 x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set ${(2 x - 1)}^{2}$ equal to $0$ .

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

Step 4.2

Solve ${(2 x - 1)}^{2} = 0$ for $x$ .

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Step 4.2.1

Set the $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 4.2.2

Solve for $x$ .

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Step 4.2.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 4.2.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 4.2.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 4.2.2.2.2

Simplify the left side.

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Step 4.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 5

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

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Step 5.1

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

$x - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6

The final solution is all the values that make $(x^{2} + 1) {(2 x - 1)}^{2} (x - 3) = 0$ true.

$x = i, - i, \frac{1}{2}, 3$

Step 7