Algebra Examples

${(10 x^{2} + 11 x)}^{2} - 15 (10 x^{2} + 11 x) - 54$

Step 1

Let $u = 10 x^{2} + 11 x$ . Substitute $u$ for all occurrences of $10 x^{2} + 11 x$ .

$u^{2} - 15 u - 54$

Step 2

Factor $u^{2} - 15 u - 54$ using the AC method.

Tap for more steps...

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 $- 54$ and whose sum is $- 15$ .

$- 18, 3$

Step 2.2

Write the factored form using these integers.

$(u - 18) (u + 3)$

Step 3

Replace all occurrences of $u$ with $10 x^{2} + 11 x$ .

$(10 x^{2} + 11 x - 18) (10 x^{2} + 11 x + 3)$

Step 4

Factor by grouping.

Tap for more steps...

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 $a \cdot c = 10 \cdot - 18 = - 180$ and whose sum is $b = 11$ .

Tap for more steps...

Step 4.1.1

Factor $11$ out of $11 x$ .

$(10 x^{2} + 11 (x) - 18) (10 x^{2} + 11 x + 3)$

Step 4.1.2

Rewrite $11$ as $- 9$ plus $20$

$(10 x^{2} + (- 9 + 20) x - 18) (10 x^{2} + 11 x + 3)$

Step 4.1.3

Apply the distributive property.

$(10 x^{2} - 9 x + 20 x - 18) (10 x^{2} + 11 x + 3)$

Step 4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.1

Group the first two terms and the last two terms.

$((10 x^{2} - 9 x) + 20 x - 18) (10 x^{2} + 11 x + 3)$

Step 4.2.2

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

$(x (10 x - 9) + 2 (10 x - 9)) (10 x^{2} + 11 x + 3)$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $10 x - 9$ .

$(10 x - 9) (x + 2) (10 x^{2} + 11 x + 3)$

Step 5

Factor.

Tap for more steps...

Step 5.1

Factor by grouping.

Tap for more steps...

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 $a \cdot c = 10 \cdot 3 = 30$ and whose sum is $b = 11$ .

Tap for more steps...

Step 5.1.1.1

Factor $11$ out of $11 x$ .

$(10 x - 9) (x + 2) (10 x^{2} + 11 (x) + 3)$

Step 5.1.1.2

Rewrite $11$ as $5$ plus $6$

$(10 x - 9) (x + 2) (10 x^{2} + (5 + 6) x + 3)$

Step 5.1.1.3

Apply the distributive property.

$(10 x - 9) (x + 2) (10 x^{2} + 5 x + 6 x + 3)$

Step 5.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.1.2.1

Group the first two terms and the last two terms.

$(10 x - 9) (x + 2) ((10 x^{2} + 5 x) + 6 x + 3)$

Step 5.1.2.2

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

$(10 x - 9) (x + 2) (5 x (2 x + 1) + 3 (2 x + 1))$

Step 5.1.3

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

$(10 x - 9) (x + 2) ((2 x + 1) (5 x + 3))$

Step 5.2

Remove unnecessary parentheses.

$(10 x - 9) (x + 2) (2 x + 1) (5 x + 3)$