Algebra Examples

$\frac{x^{2} + 6 x - 7}{x^{4} + 8 x^{3} + 7 x^{2}} \cdot 3 x^{2}$

Step 1

Rewrite using the commutative property of multiplication.

$3 \frac{x^{2} + 6 x - 7}{x^{4} + 8 x^{3} + 7 x^{2}} x^{2}$

Step 2

Factor $x^{2} + 6 x - 7$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 2.2

Write the factored form using these integers.

$3 \frac{(x - 1) (x + 7)}{x^{4} + 8 x^{3} + 7 x^{2}} x^{2}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $x^{2}$ out of $x^{4} + 8 x^{3} + 7 x^{2}$ .

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Step 3.1.1

Factor $x^{2}$ out of $x^{4}$ .

$3 \frac{(x - 1) (x + 7)}{x^{2} x^{2} + 8 x^{3} + 7 x^{2}} x^{2}$

Step 3.1.2

Factor $x^{2}$ out of $8 x^{3}$ .

$3 \frac{(x - 1) (x + 7)}{x^{2} x^{2} + x^{2} (8 x) + 7 x^{2}} x^{2}$

Step 3.1.3

Factor $x^{2}$ out of $7 x^{2}$ .

$3 \frac{(x - 1) (x + 7)}{x^{2} x^{2} + x^{2} (8 x) + x^{2} \cdot 7} x^{2}$

Step 3.1.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (8 x)$ .

$3 \frac{(x - 1) (x + 7)}{x^{2} (x^{2} + 8 x) + x^{2} \cdot 7} x^{2}$

Step 3.1.5

Factor $x^{2}$ out of $x^{2} (x^{2} + 8 x) + x^{2} \cdot 7$ .

$3 \frac{(x - 1) (x + 7)}{x^{2} (x^{2} + 8 x + 7)} x^{2}$

Step 3.2

Factor $x^{2} + 8 x + 7$ using the AC method.

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Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $7$ and whose sum is $8$ .

$1, 7$

Step 3.2.2

Write the factored form using these integers.

$3 \frac{(x - 1) (x + 7)}{x^{2} ((x + 1) (x + 7))} x^{2}$

$3 \frac{(x - 1) (x + 7)}{x^{2} (x + 1) (x + 7)} x^{2}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 7$ .

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Step 4.1.1

Cancel the common factor.

$3 \frac{(x - 1) (x + 7)}{x^{2} (x + 1) (x + 7)} x^{2}$

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Combine $3$ and $\frac{x - 1}{x^{2} (x + 1)}$ .

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

Step 4.3

Cancel the common factor of $x^{2}$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$\frac{3 (x - 1)}{x + 1}$