Algebra Examples

$- x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8$

Step 1

Write $- x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8$ as an equation.

$y = - x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $- x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8 = 0$ .

$- x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8 = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $- x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8$ using the rational roots test.

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Step 2.2.2.1.1

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.

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

$q = \pm 1$

Step 2.2.2.1.2

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

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

Step 2.2.2.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 2.2.2.1.3.1

Substitute $2$ into the polynomial.

$- 2^{4} + 9 \cdot 2^{3} - 22 \cdot 2^{2} + 12 \cdot 2 + 8$

Step 2.2.2.1.3.2

Raise $2$ to the power of $4$ .

$- 1 \cdot 16 + 9 \cdot 2^{3} - 22 \cdot 2^{2} + 12 \cdot 2 + 8$

Step 2.2.2.1.3.3

Multiply $- 1$ by $16$ .

$- 16 + 9 \cdot 2^{3} - 22 \cdot 2^{2} + 12 \cdot 2 + 8$

Step 2.2.2.1.3.4

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

$- 16 + 9 \cdot 8 - 22 \cdot 2^{2} + 12 \cdot 2 + 8$

Step 2.2.2.1.3.5

Multiply $9$ by $8$ .

$- 16 + 72 - 22 \cdot 2^{2} + 12 \cdot 2 + 8$

Step 2.2.2.1.3.6

Add $- 16$ and $72$ .

$56 - 22 \cdot 2^{2} + 12 \cdot 2 + 8$

Step 2.2.2.1.3.7

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

$56 - 22 \cdot 4 + 12 \cdot 2 + 8$

Step 2.2.2.1.3.8

Multiply $- 22$ by $4$ .

$56 - 88 + 12 \cdot 2 + 8$

Step 2.2.2.1.3.9

Subtract $88$ from $56$ .

$- 32 + 12 \cdot 2 + 8$

Step 2.2.2.1.3.10

Multiply $12$ by $2$ .

$- 32 + 24 + 8$

Step 2.2.2.1.3.11

Add $- 32$ and $24$ .

$- 8 + 8$

Step 2.2.2.1.3.12

Add $- 8$ and $8$ .

$0$

Step 2.2.2.1.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8}{x - 2}$

Step 2.2.2.1.5

Divide $- x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8$ by $x - 2$ .

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Step 2.2.2.1.5.1

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

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

Step 2.2.2.1.5.2

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

				-	$x^{3}$
	$x$	-	$2$	-	$x^{4}$	+	$9 x^{3}$	-	$22 x^{2}$	+	$12 x$	+	$8$

Step 2.2.2.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

Step 2.2.2.1.5.4

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

$x^{3}$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

Step 2.2.2.1.5.5

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

$x^{3}$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

Step 2.2.2.1.5.6

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

$x^{3}$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

Step 2.2.2.1.5.7

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

Step 2.2.2.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

Step 2.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} - 14 x^{2}$

$x^{3}$

$7 x^{2}$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

Step 2.2.2.1.5.10

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

$x^{3}$

$7 x^{2}$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

Step 2.2.2.1.5.11

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

$x^{3}$

$7 x^{2}$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

Step 2.2.2.1.5.12

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$8 x$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

Step 2.2.2.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$8 x$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$16 x$

Step 2.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} + 16 x$

$x^{3}$

$7 x^{2}$

$8 x$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$16 x$

Step 2.2.2.1.5.15

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

$x^{3}$

$7 x^{2}$

$8 x$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$16 x$

$4 x$

Step 2.2.2.1.5.16

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

$x^{3}$

$7 x^{2}$

$8 x$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$16 x$

$4 x$

$8$

Step 2.2.2.1.5.17

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $x$ .

$x^{3}$

$7 x^{2}$

$8 x$

$4$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$16 x$

$4 x$

$8$

Step 2.2.2.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$7 x^{2}$

$8 x$

$4$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$16 x$

$4 x$

$8$

$4 x$

$8$

Step 2.2.2.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x + 8$

$x^{3}$

$7 x^{2}$

$8 x$

$4$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$16 x$

$4 x$

$8$

$4 x$

$8$

Step 2.2.2.1.5.20

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

$x^{3}$

$7 x^{2}$

$8 x$

$4$

$x$

$2$

$x^{4}$

$9 x^{3}$

$22 x^{2}$

$12 x$

$8$

$x^{4}$

$2 x^{3}$

$7 x^{3}$

$22 x^{2}$

$7 x^{3}$

$14 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$16 x$

$4 x$

$8$

$4 x$

$8$

$0$

Step 2.2.2.1.5.21

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

$- x^{3} + 7 x^{2} - 8 x - 4$

Step 2.2.2.1.6

Write $- x^{4} + 9 x^{3} - 22 x^{2} + 12 x + 8$ as a set of factors.

$(x - 2) (- x^{3} + 7 x^{2} - 8 x - 4) = 0$

Step 2.2.2.2

Factor $- x^{3} + 7 x^{2} - 8 x - 4$ using the rational roots test.

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Step 2.2.2.2.1

Factor $- x^{3} + 7 x^{2} - 8 x - 4$ using the rational roots test.

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Step 2.2.2.2.1.1

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.

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

$q = \pm 1$

Step 2.2.2.2.1.2

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

$\pm 1, \pm 4, \pm 2$

Step 2.2.2.2.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 2.2.2.2.1.3.1

Substitute $2$ into the polynomial.

$- 2^{3} + 7 \cdot 2^{2} - 8 \cdot 2 - 4$

Step 2.2.2.2.1.3.2

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

$- 1 \cdot 8 + 7 \cdot 2^{2} - 8 \cdot 2 - 4$

Step 2.2.2.2.1.3.3

Multiply $- 1$ by $8$ .

$- 8 + 7 \cdot 2^{2} - 8 \cdot 2 - 4$

Step 2.2.2.2.1.3.4

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

$- 8 + 7 \cdot 4 - 8 \cdot 2 - 4$

Step 2.2.2.2.1.3.5

Multiply $7$ by $4$ .

$- 8 + 28 - 8 \cdot 2 - 4$

Step 2.2.2.2.1.3.6

Add $- 8$ and $28$ .

$20 - 8 \cdot 2 - 4$

Step 2.2.2.2.1.3.7

Multiply $- 8$ by $2$ .

$20 - 16 - 4$

Step 2.2.2.2.1.3.8

Subtract $16$ from $20$ .

$4 - 4$

Step 2.2.2.2.1.3.9

Subtract $4$ from $4$ .

$0$

Step 2.2.2.2.1.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- x^{3} + 7 x^{2} - 8 x - 4}{x - 2}$

Step 2.2.2.2.1.5

Divide $- x^{3} + 7 x^{2} - 8 x - 4$ by $x - 2$ .

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Step 2.2.2.2.1.5.1

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

$x$

$2$

$x^{3}$

$7 x^{2}$

$8 x$

$4$

Step 2.2.2.2.1.5.2

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

				-	$x^{2}$
	$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$

Step 2.2.2.2.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			-	$x^{3}$	+	$2 x^{2}$

Step 2.2.2.2.1.5.4

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

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$

Step 2.2.2.2.1.5.5

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

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$

Step 2.2.2.2.1.5.6

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

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$

Step 2.2.2.2.1.5.7

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$5 x$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$

Step 2.2.2.2.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$5 x$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$
					+	$5 x^{2}$	-	$10 x$

Step 2.2.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} - 10 x$

			-	$x^{2}$	+	$5 x$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$
					-	$5 x^{2}$	+	$10 x$

Step 2.2.2.2.1.5.10

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

			-	$x^{2}$	+	$5 x$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$
					-	$5 x^{2}$	+	$10 x$
							+	$2 x$

Step 2.2.2.2.1.5.11

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

			-	$x^{2}$	+	$5 x$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$
					-	$5 x^{2}$	+	$10 x$
							+	$2 x$	-	$4$

Step 2.2.2.2.1.5.12

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

			-	$x^{2}$	+	$5 x$	+	$2$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$
					-	$5 x^{2}$	+	$10 x$
							+	$2 x$	-	$4$

Step 2.2.2.2.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$5 x$	+	$2$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$
					-	$5 x^{2}$	+	$10 x$
							+	$2 x$	-	$4$
							+	$2 x$	-	$4$

Step 2.2.2.2.1.5.14

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

			-	$x^{2}$	+	$5 x$	+	$2$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$
					-	$5 x^{2}$	+	$10 x$
							+	$2 x$	-	$4$
							-	$2 x$	+	$4$

Step 2.2.2.2.1.5.15

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

			-	$x^{2}$	+	$5 x$	+	$2$
$x$	-	$2$	-	$x^{3}$	+	$7 x^{2}$	-	$8 x$	-	$4$
			+	$x^{3}$	-	$2 x^{2}$
					+	$5 x^{2}$	-	$8 x$
					-	$5 x^{2}$	+	$10 x$
							+	$2 x$	-	$4$
							-	$2 x$	+	$4$
										$0$

Step 2.2.2.2.1.5.16

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

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

Step 2.2.2.2.1.6

Write $- x^{3} + 7 x^{2} - 8 x - 4$ as a set of factors.

$(x - 2) ((x - 2) (- x^{2} + 5 x + 2)) = 0$

Step 2.2.2.2.2

Remove unnecessary parentheses.

$(x - 2) (x - 2) (- x^{2} + 5 x + 2) = 0$

Step 2.2.2.3

Combine like factors.

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Step 2.2.2.3.1

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

$(x - 2) (x - 2) (- x^{2} + 5 x + 2) = 0$

Step 2.2.2.3.2

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

$(x - 2) (x - 2) (- x^{2} + 5 x + 2) = 0$

Step 2.2.2.3.3

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

${(x - 2)}^{1 + 1} (- x^{2} + 5 x + 2) = 0$

Step 2.2.2.3.4

Add $1$ and $1$ .

${(x - 2)}^{2} (- x^{2} + 5 x + 2) = 0$

Step 2.2.3

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

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

$- x^{2} + 5 x + 2 = 0$

Step 2.2.4

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

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

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Step 2.2.4.2.1

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

$x - 2 = 0$

Step 2.2.4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.2.5

Set $- x^{2} + 5 x + 2$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $- x^{2} + 5 x + 2$ equal to $0$ .

$- x^{2} + 5 x + 2 = 0$

Step 2.2.5.2

Solve $- x^{2} + 5 x + 2 = 0$ for $x$ .

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Step 2.2.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2.5.2.2

Substitute the values $a = - 1$ , $b = 5$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (- 1 \cdot 2)}}{2 \cdot - 1}$

Step 2.2.5.2.3

Simplify.

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Step 2.2.5.2.3.1

Simplify the numerator.

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Step 2.2.5.2.3.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot 2}}{2 \cdot - 1}$

Step 2.2.5.2.3.1.2

Multiply $- 4 \cdot - 1 \cdot 2$ .

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Step 2.2.5.2.3.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot 2}}{2 \cdot - 1}$

Step 2.2.5.2.3.1.2.2

Multiply $4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 + 8}}{2 \cdot - 1}$

Step 2.2.5.2.3.1.3

Add $25$ and $8$ .

$x = \frac{- 5 \pm \sqrt{33}}{2 \cdot - 1}$

Step 2.2.5.2.3.2

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{33}}{- 2}$

Step 2.2.5.2.3.3

Simplify $\frac{- 5 \pm \sqrt{33}}{- 2}$ .

$x = \frac{5 \pm \sqrt{33}}{2}$

Step 2.2.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.2.5.2.4.1

Simplify the numerator.

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Step 2.2.5.2.4.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot 2}}{2 \cdot - 1}$

Step 2.2.5.2.4.1.2

Multiply $- 4 \cdot - 1 \cdot 2$ .

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Step 2.2.5.2.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot 2}}{2 \cdot - 1}$

Step 2.2.5.2.4.1.2.2

Multiply $4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 + 8}}{2 \cdot - 1}$

Step 2.2.5.2.4.1.3

Add $25$ and $8$ .

$x = \frac{- 5 \pm \sqrt{33}}{2 \cdot - 1}$

Step 2.2.5.2.4.2

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{33}}{- 2}$

Step 2.2.5.2.4.3

Simplify $\frac{- 5 \pm \sqrt{33}}{- 2}$ .

$x = \frac{5 \pm \sqrt{33}}{2}$

Step 2.2.5.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{33}}{2}$

Step 2.2.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.2.5.2.5.1

Simplify the numerator.

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Step 2.2.5.2.5.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot 2}}{2 \cdot - 1}$

Step 2.2.5.2.5.1.2

Multiply $- 4 \cdot - 1 \cdot 2$ .

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Step 2.2.5.2.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot 2}}{2 \cdot - 1}$

Step 2.2.5.2.5.1.2.2

Multiply $4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 + 8}}{2 \cdot - 1}$

Step 2.2.5.2.5.1.3

Add $25$ and $8$ .

$x = \frac{- 5 \pm \sqrt{33}}{2 \cdot - 1}$

Step 2.2.5.2.5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{33}}{- 2}$

Step 2.2.5.2.5.3

Simplify $\frac{- 5 \pm \sqrt{33}}{- 2}$ .

$x = \frac{5 \pm \sqrt{33}}{2}$

Step 2.2.5.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{33}}{2}$

Step 2.2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{33}}{2}, \frac{5 - \sqrt{33}}{2}$

Step 2.2.6

The final solution is all the values that make ${(x - 2)}^{2} (- x^{2} + 5 x + 2) = 0$ true.

$x = 2, \frac{5 + \sqrt{33}}{2}, \frac{5 - \sqrt{33}}{2}$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (\frac{5 + \sqrt{33}}{2}, 0), (\frac{5 - \sqrt{33}}{2}, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - {(0)}^{4} + 9 {(0)}^{3} - 22 {(0)}^{2} + 12 (0) + 8$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = - 0^{4} + 9 {(0)}^{3} - 22 {(0)}^{2} + 12 (0) + 8$

Step 3.2.2

Remove parentheses.

$y = - 0^{4} + 9 \cdot 0^{3} - 22 {(0)}^{2} + 12 (0) + 8$

Step 3.2.3

Remove parentheses.

$y = - 0^{4} + 9 \cdot 0^{3} - 22 \cdot 0^{2} + 12 (0) + 8$

Step 3.2.4

Remove parentheses.

$y = - {(0)}^{4} + 9 {(0)}^{3} - 22 {(0)}^{2} + 12 (0) + 8$

Step 3.2.5

Simplify $- {(0)}^{4} + 9 {(0)}^{3} - 22 {(0)}^{2} + 12 (0) + 8$ .

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Step 3.2.5.1

Simplify each term.

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Step 3.2.5.1.1

Raising $0$ to any positive power yields $0$ .

$y = - 0 + 9 {(0)}^{3} - 22 {(0)}^{2} + 12 (0) + 8$

Step 3.2.5.1.2

Multiply $- 1$ by $0$ .

$y = 0 + 9 {(0)}^{3} - 22 {(0)}^{2} + 12 (0) + 8$

Step 3.2.5.1.3

Raising $0$ to any positive power yields $0$ .

$y = 0 + 9 \cdot 0 - 22 {(0)}^{2} + 12 (0) + 8$

Step 3.2.5.1.4

Multiply $9$ by $0$ .

$y = 0 + 0 - 22 {(0)}^{2} + 12 (0) + 8$

Step 3.2.5.1.5

Raising $0$ to any positive power yields $0$ .

$y = 0 + 0 - 22 \cdot 0 + 12 (0) + 8$

Step 3.2.5.1.6

Multiply $- 22$ by $0$ .

$y = 0 + 0 + 0 + 12 (0) + 8$

Step 3.2.5.1.7

Multiply $12$ by $0$ .

$y = 0 + 0 + 0 + 0 + 8$

Step 3.2.5.2

Simplify by adding numbers.

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Step 3.2.5.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 0 + 8$

Step 3.2.5.2.2

Add $0$ and $0$ .

$y = 0 + 0 + 8$

Step 3.2.5.2.3

Add $0$ and $0$ .

$y = 0 + 8$

Step 3.2.5.2.4

Add $0$ and $8$ .

$y = 8$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 8)$

Step 4

List the intersections.

x-intercept(s): $(2, 0), (\frac{5 + \sqrt{33}}{2}, 0), (\frac{5 - \sqrt{33}}{2}, 0)$

y-intercept(s): $(0, 8)$

Step 5