Factoring Binomials

Factor the polynomial: (x² - 5x + 6)

• (x - 2)(x - 2)

• (x - 3)(x - 3)

• (x - 2)(x - 3)

• (x - 1)(x - 6)

Answer explanation

To factor the polynomial x² - 5x + 6, we look for two numbers that multiply to 6 and add to -5. The numbers -2 and -3 fit this, so we can write it as (x - 2)(x - 3). Thus, the correct choice is (x - 2)(x - 3).

Factor the polynomial: (x² + 7x + 12)

• (x + 4)(x + 5)

• (x + 1)(x + 12)

• (x + 2)(x + 6)

• (x + 3)(x + 4)

Answer explanation

To factor x² + 7x + 12, we look for two numbers that multiply to 12 and add to 7. The numbers 3 and 4 fit this, so the polynomial factors to (x + 3)(x + 4). Thus, the correct answer is (x + 3)(x + 4).

Factor the polynomial: (x² - 4x - 12)

• (x - 3)(x + 4)

• (x - 6)(x + 2)

• (x - 2)(x + 6)

• (x - 4)(x + 3)

Answer explanation

To factor x² - 4x - 12, we look for two numbers that multiply to -12 and add to -4. The numbers -6 and 2 fit this, so the polynomial factors to (x - 6)(x + 2). Thus, the correct answer is (x - 6)(x + 2).

Factor the polynomial: (x² + 8x + 16)

• (x + 5)^2

• (x + 2)^2

• (x + 4)^2

• (x + 3)^2

Answer explanation

To factor the polynomial x^2 + 8x + 16, we look for two numbers that multiply to 16 and add to 8. The numbers 4 and 4 fit this, so we can write it as (x + 4)(x + 4) or (x + 4)^2.

Factor the polynomial: (x² - 16x + 64)

• (x - 6)^2

• (x - 4)^2

• (x - 8)^2

• (x - 7)^2

Answer explanation

To factor the polynomial x² - 16x + 64, we look for two identical factors that add to -16 and multiply to 64. The expression can be rewritten as (x - 8)(x - 8), which simplifies to (x - 8)².

Factor the polynomial: (x² - 6x)

• x(x - 4)

• x(x - 3)

• x(x - 6)

• x(x - 2)

Answer explanation

To factor the polynomial x^2 - 6x, we can factor out the common term x, resulting in x(x - 6). Thus, the correct answer is x(x - 6).

Factor the polynomial: (3x² + 12x + 12)

• 3(x + 1)^2

• 3(x + 2)^2

• 3(x + 3)^2

• 3(x + 4)^2

Answer explanation

To factor 3x^2 + 12x + 12, first factor out 3: 3(x^2 + 4x + 4). Then, recognize that x^2 + 4x + 4 is a perfect square: (x + 2)^2. Thus, the factored form is 3(x + 2)^2.

Factor the polynomial: (2x² - 8x + 6)

• 2(x - 1)(x - 4)

• 2(x - 1)(x - 3)

• 2(x - 2)(x - 2)

• 2(x - 2)(x - 3)

Answer explanation

To factor 2x^2 - 8x + 6, first factor out 2: 2(x^2 - 4x + 3). Then, factor the quadratic: (x - 1)(x - 3). Thus, the complete factorization is 2(x - 1)(x - 3), which matches the correct answer.

Factor the polynomial: (x² - 10x + 25)

• (x - 6)^2

• (x - 4)^2

• (x - 3)^2

• (x - 5)^2

Answer explanation

The polynomial x² - 10x + 25 can be factored as (x - 5)(x - 5) or (x - 5)², since it is a perfect square trinomial. Thus, the correct answer is (x - 5)².

Multiply the binomials: (x + 2)(x + 3)

• x^2 + 3x + 4

• x^2 + 5x + 6

• x^2 + 4x + 5

• x^2 + 6x + 7

Answer explanation

To multiply the binomials (x + 2)(x + 3), use the distributive property: x*x + x*3 + 2*x + 2*3 = x^2 + 3x + 2x + 6 = x^2 + 5x + 6. Thus, the correct answer is x^2 + 5x + 6.