Factorize the following algebraic expressions: 1. a2 - a − b − b2 2. a2 - 2a + 2b − b2 3. x2 + 5x + 6 4. x2…

Question

Factorize the following algebraic expressions: 1. a2 - a − b − b2 2. a2 - 2a + 2b − b2 3. x2 + 5x + 6 4. x2 + 5x − 6

ScanSolve AI Tutor · Accepted Answer

Here are the factorizations for each expression:

$a^2 - a - b - b^2$

Rearrange the terms to group $a^2$ with $b^2$ .
$$ a^2 - b^2 - a - b $$
Factor the difference of squares $a^2 - b^2$ .
$$ (a-b)(a+b) - a - b $$
Factor out $-1$ from the last two terms.
$$ (a-b)(a+b) - (a+b) $$
Factor out the common term $(a+b)$ .
$$ (a+b)(a-b-1) $$ The factorization is $\boxed{(a+b)(a-b-1)}$.
2. $a^2 - 2a + 2b - b^2$
Rearrange the terms to group $a^2$ with $b^2$ .
$$ a^2 - b^2 - 2a + 2b $$
Factor the difference of squares $a^2 - b^2$ .
$$ (a-b)(a+b) - 2a + 2b $$
Factor out $-2$ from the last two terms.
$$ (a-b)(a+b) - 2(a-b) $$
Factor out the common term $(a-b)$ .
$$ (a-b)(a+b-2) $$ The factorization is $\boxed{(a-b)(a+b-2)}$.
3. $x^2 + 5x + 6$
Find two numbers that multiply to 6 and add up to 5.
These numbers are 2 and 3.
Rewrite the middle term using these numbers.
$$ x^2 + 2x + 3x + 6 $$
Group terms and factor.
$$ x(x+2) + 3(x+2) $$
Factor out the common term $(x+2)$ .
$$ (x+2)(x+3) $$ The factorization is $\boxed{(x+2)(x+3)}$.
4. $x^2 + 5x - 6$
Find two numbers that multiply to -6 and add up to 5.
These numbers are 6 and -1.
Rewrite the middle term using these numbers.
$$ x^2 + 6x - x - 6 $$
Group terms and factor.
$$ x(x+6) - 1(x+6) $$
Factor out the common term $(x+6)$ .
$$ (x+6)(x-1) $$ The factorization is $\boxed{(x+6)(x-1)}$.
5. $x^2 + 3x + 2$
Find two numbers that multiply to 2 and add up to 3.
These numbers are 1 and 2.
Rewrite the middle term using these numbers.
$$ x^2 + x + 2x + 2 $$
Group terms and factor.
$$ x(x+1) + 2(x+1) $$

Factor out the common term $(x+1)$ .

$$
(x+1)(x+2)
$$
The factorization is $\boxed{(x+1)(x+2)}$.

Factorize the following algebraic expressions: 1. a2 - a - b - b2 2. a2 - 2a + 2b - b2 3. x2 + 5x + 6 4. x2 + 5x - 6

Step-by-step solution

Rearrange the terms to group $a^2$ with $b^2$ .

Factor the difference of squares $a^2 - b^2$ .

Factor out $-1$ from the last two terms.

Factor out the common term $(a+b)$ .

Rearrange the terms to group $a^2$ with $b^2$ .

Factor the difference of squares $a^2 - b^2$ .

Factor out $-2$ from the last two terms.

Factor out the common term $(a-b)$ .

Find two numbers that multiply to 6 and add up to 5.

Rewrite the middle term using these numbers.

Group terms and factor.

Factor out the common term $(x+2)$ .

Find two numbers that multiply to -6 and add up to 5.

Rewrite the middle term using these numbers.

Group terms and factor.

Factor out the common term $(x+6)$ .

Find two numbers that multiply to 2 and add up to 3.

Rewrite the middle term using these numbers.

Group terms and factor.

Factor out the common term $(x+1)$ .

Still stuck on this one?

More Mathematics Questions

Factorize the following algebraic expressions: 1. a2 - a - b - b2 2. a2 - 2a + 2b - b2 3. x2 + 5x + 6 4. x2 + 5x - 6

Step-by-step solution

Rearrange the terms to group a2a^2a2 with b2b^2b2.

Factor the difference of squares a2−b2a^2 - b^2a2−b2.

Factor out −1-1−1 from the last two terms.

Factor out the common term (a+b)(a+b)(a+b).

Rearrange the terms to group a2a^2a2 with b2b^2b2.

Factor the difference of squares a2−b2a^2 - b^2a2−b2.

Factor out −2-2−2 from the last two terms.

Factor out the common term (a−b)(a-b)(a−b).

Find two numbers that multiply to 6 and add up to 5.

Rewrite the middle term using these numbers.

Group terms and factor.

Factor out the common term (x+2)(x+2)(x+2).

Find two numbers that multiply to -6 and add up to 5.

Rewrite the middle term using these numbers.

Group terms and factor.

Factor out the common term (x+6)(x+6)(x+6).

Find two numbers that multiply to 2 and add up to 3.

Rewrite the middle term using these numbers.

Group terms and factor.

Factor out the common term (x+1)(x+1)(x+1).

Still stuck on this one?

More Mathematics Questions

Rearrange the terms to group $a^2$ with $b^2$ .

Factor the difference of squares $a^2 - b^2$ .

Factor out $-1$ from the last two terms.

Factor out the common term $(a+b)$ .

Rearrange the terms to group $a^2$ with $b^2$ .

Factor the difference of squares $a^2 - b^2$ .

Factor out $-2$ from the last two terms.

Factor out the common term $(a-b)$ .

Factor out the common term $(x+2)$ .

Factor out the common term $(x+6)$ .

Factor out the common term $(x+1)$ .