Algebra Examples

(6 z^{4} + 3 z^{2} - 9) {(3 z^{2} - 6)}^{- 1}

$(6 z^{4} + 3 z^{2} - 9) {(3 z^{2} - 6)}^{- 1}$

Step 1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 z^{2} - 6}

$(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 z^{2} - 6}$

Step 2

Simplify terms.

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Step 2.1

Factor

3

$3$ out of

3 z^{2} - 6

$3 z^{2} - 6$ .

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Step 2.1.1

Factor

3

$3$ out of

3 z^{2}

$3 z^{2}$ .

(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 (z^{2}) - 6}

$(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 (z^{2}) - 6}$

Step 2.1.2

Factor

3

$3$ out of

- 6

$- 6$ .

(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 z^{2} + 3 \cdot - 2}

$(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 z^{2} + 3 \cdot - 2}$

Step 2.1.3

Factor

3

$3$ out of

3 z^{2} + 3 \cdot - 2

$3 z^{2} + 3 \cdot - 2$ .

(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 (z^{2} - 2)}

$(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 (z^{2} - 2)}$

(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 (z^{2} - 2)}

$(6 z^{4} + 3 z^{2} - 9) \frac{1}{3 (z^{2} - 2)}$

Step 2.2

Multiply

6 z^{4} + 3 z^{2} - 9

$6 z^{4} + 3 z^{2} - 9$ by

\frac{1}{3 (z^{2} - 2)}

$\frac{1}{3 (z^{2} - 2)}$ .

\frac{6 z^{4} + 3 z^{2} - 9}{3 (z^{2} - 2)}

$\frac{6 z^{4} + 3 z^{2} - 9}{3 (z^{2} - 2)}$

Step 2.3

Cancel the common factor of

6 z^{4} + 3 z^{2} - 9

$6 z^{4} + 3 z^{2} - 9$ and

3

$3$ .

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Step 2.3.1

Factor

3

$3$ out of

6 z^{4}

$6 z^{4}$ .

\frac{3 (2 z^{4}) + 3 z^{2} - 9}{3 (z^{2} - 2)}

$\frac{3 (2 z^{4}) + 3 z^{2} - 9}{3 (z^{2} - 2)}$

Step 2.3.2

Factor

3

$3$ out of

3 z^{2}

$3 z^{2}$ .

\frac{3 (2 z^{4}) + 3 (z^{2}) - 9}{3 (z^{2} - 2)}

$\frac{3 (2 z^{4}) + 3 (z^{2}) - 9}{3 (z^{2} - 2)}$

Step 2.3.3

Factor

$3$ out of

$3 (2 z^{4}) + 3 (z^{2})$ .

$\frac{3 (2 z^{4} + z^{2}) - 9}{3 (z^{2} - 2)}$

Step 2.3.4

Factor

$3$ out of

$- 9$ .

$\frac{3 (2 z^{4} + z^{2}) + 3 \cdot - 3}{3 (z^{2} - 2)}$

Step 2.3.5

Factor

$3$ out of

$3 (2 z^{4} + z^{2}) + 3 (- 3)$ .

$\frac{3 (2 z^{4} + z^{2} - 3)}{3 (z^{2} - 2)}$

Step 2.3.6

Cancel the common factors.

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Step 2.3.6.1

Cancel the common factor.

$\frac{3 (2 z^{4} + z^{2} - 3)}{3 (z^{2} - 2)}$

Step 2.3.6.2

Rewrite the expression.

$\frac{2 z^{4} + z^{2} - 3}{z^{2} - 2}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite

$z^{4}$ as

${(z^{2})}^{2}$ .

$\frac{2 {(z^{2})}^{2} + z^{2} - 3}{z^{2} - 2}$

Step 3.2

Let

$u = z^{2}$ . Substitute

$u$ for all occurrences of

$z^{2}$ .

$\frac{2 u^{2} + u - 3}{z^{2} - 2}$

Step 3.3

Factor by grouping.

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Step 3.3.1

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

$b = 1$ .

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Step 3.3.1.1

Multiply by

$1$ .

$\frac{2 u^{2} + 1 u - 3}{z^{2} - 2}$

Step 3.3.1.2

Rewrite

$1$ as

$- 2$ plus

$3$

$\frac{2 u^{2} + (- 2 + 3) u - 3}{z^{2} - 2}$

Step 3.3.1.3

Apply the distributive property.

$\frac{2 u^{2} - 2 u + 3 u - 3}{z^{2} - 2}$

Step 3.3.2

Factor out the greatest common factor from each group.

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Step 3.3.2.1

Group the first two terms and the last two terms.

$\frac{(2 u^{2} - 2 u) + 3 u - 3}{z^{2} - 2}$

Step 3.3.2.2

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

$\frac{2 u (u - 1) + 3 (u - 1)}{z^{2} - 2}$

Step 3.3.3

Factor the polynomial by factoring out the greatest common factor,

$u - 1$ .

$\frac{(u - 1) (2 u + 3)}{z^{2} - 2}$

Step 3.4

Replace all occurrences of

$u$ with

$z^{2}$ .

$\frac{(z^{2} - 1) (2 z^{2} + 3)}{z^{2} - 2}$

Step 3.5

Rewrite

$1$ as

$1^{2}$ .

$\frac{(z^{2} - 1^{2}) (2 z^{2} + 3)}{z^{2} - 2}$

Step 3.6

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = z$ and

$b = 1$ .

$\frac{(z + 1) (z - 1) (2 z^{2} + 3)}{z^{2} - 2}$