Linear Algebra Examples

$\frac{4 M^{2} + 7 M - 2}{2 M^{2} - 8} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 1

Factor by grouping.

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Step 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 $a \cdot c = 4 \cdot - 2 = - 8$ and whose sum is $b = 7$ .

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Step 1.1.1

Factor $7$ out of $7 M$ .

$\frac{4 M^{2} + 7 (M) - 2}{2 M^{2} - 8} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 1.1.2

Rewrite $7$ as $- 1$ plus $8$

$\frac{4 M^{2} + (- 1 + 8) M - 2}{2 M^{2} - 8} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 1.1.3

Apply the distributive property.

$\frac{4 M^{2} - 1 M + 8 M - 2}{2 M^{2} - 8} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(4 M^{2} - 1 M) + 8 M - 2}{2 M^{2} - 8} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 1.2.2

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

$\frac{M (4 M - 1) + 2 (4 M - 1)}{2 M^{2} - 8} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $4 M - 1$ .

$\frac{(4 M - 1) (M + 2)}{2 M^{2} - 8} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2$ out of $2 M^{2} - 8$ .

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Step 2.1.1

Factor $2$ out of $2 M^{2}$ .

$\frac{(4 M - 1) (M + 2)}{2 (M^{2}) - 8} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 2.1.2

Factor $2$ out of $- 8$ .

$\frac{(4 M - 1) (M + 2)}{2 M^{2} + 2 \cdot - 4} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 2.1.3

Factor $2$ out of $2 M^{2} + 2 \cdot - 4$ .

$\frac{(4 M - 1) (M + 2)}{2 (M^{2} - 4)} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 2.2

Rewrite $4$ as $2^{2}$ .

$\frac{(4 M - 1) (M + 2)}{2 (M^{2} - 2^{2})} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = M$ and $b = 2$ .

$\frac{(4 M - 1) (M + 2)}{2 (M + 2) (M - 2)} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 3

Cancel the common factor of $M + 2$ .

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Step 3.1

Cancel the common factor.

$\frac{(4 M - 1) (M + 2)}{2 (M + 2) (M - 2)} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 3.2

Rewrite the expression.

$\frac{4 M - 1}{2 (M - 2)} \cdot {(\frac{1}{M - 2})}^{- 1}$

Step 4

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{4 M - 1}{2 (M - 2)} \cdot (M - 2)$

Step 5

Cancel the common factor of $M - 2$ .

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Step 5.1

Factor $M - 2$ out of $2 (M - 2)$ .

$\frac{4 M - 1}{(M - 2) \cdot 2} \cdot (M - 2)$

Step 5.2

Cancel the common factor.

$\frac{4 M - 1}{(M - 2) \cdot 2} \cdot (M - 2)$

Step 5.3

Rewrite the expression.

$\frac{4 M - 1}{2}$