Algebra Examples

${(6 x^{2} + 7 x)}^{2} - 4 (6 x^{2} + 7 x) - 5$

Step 1

Let $u = 6 x^{2} + 7 x$ . Substitute $u$ for all occurrences of $6 x^{2} + 7 x$ .

$u^{2} - 4 u - 5$

Step 2

Factor $u^{2} - 4 u - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 2.2

Write the factored form using these integers.

$(u - 5) (u + 1)$

Step 3

Replace all occurrences of $u$ with $6 x^{2} + 7 x$ .

$(6 x^{2} + 7 x - 5) (6 x^{2} + 7 x + 1)$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 6 \cdot - 5 = - 30$ and whose sum is $b = 7$ .

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Step 4.1.1

Factor $7$ out of $7 x$ .

$(6 x^{2} + 7 (x) - 5) (6 x^{2} + 7 x + 1)$

Step 4.1.2

Rewrite $7$ as $- 3$ plus $10$

$(6 x^{2} + (- 3 + 10) x - 5) (6 x^{2} + 7 x + 1)$

Step 4.1.3

Apply the distributive property.

$(6 x^{2} - 3 x + 10 x - 5) (6 x^{2} + 7 x + 1)$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$((6 x^{2} - 3 x) + 10 x - 5) (6 x^{2} + 7 x + 1)$

Step 4.2.2

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

$(3 x (2 x - 1) + 5 (2 x - 1)) (6 x^{2} + 7 x + 1)$

Step 4.3

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

$(2 x - 1) (3 x + 5) (6 x^{2} + 7 x + 1)$

Step 5

Factor.

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Step 5.1

Factor by grouping.

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Step 5.1.1

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 $a \cdot c = 6 \cdot 1 = 6$ and whose sum is $b = 7$ .

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Step 5.1.1.1

Factor $7$ out of $7 x$ .

$(2 x - 1) (3 x + 5) (6 x^{2} + 7 (x) + 1)$

Step 5.1.1.2

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

$(2 x - 1) (3 x + 5) (6 x^{2} + (1 + 6) x + 1)$

Step 5.1.1.3

Apply the distributive property.

$(2 x - 1) (3 x + 5) (6 x^{2} + 1 x + 6 x + 1)$

Step 5.1.1.4

Multiply $x$ by $1$ .

$(2 x - 1) (3 x + 5) (6 x^{2} + x + 6 x + 1)$

Step 5.1.2

Factor out the greatest common factor from each group.

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Step 5.1.2.1

Group the first two terms and the last two terms.

$(2 x - 1) (3 x + 5) ((6 x^{2} + x) + 6 x + 1)$

Step 5.1.2.2

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

$(2 x - 1) (3 x + 5) (x (6 x + 1) + 1 (6 x + 1))$

Step 5.1.3

Factor the polynomial by factoring out the greatest common factor, $6 x + 1$ .

$(2 x - 1) (3 x + 5) ((6 x + 1) (x + 1))$

Step 5.2

Remove unnecessary parentheses.

$(2 x - 1) (3 x + 5) (6 x + 1) (x + 1)$