Rational Expressions and Factoring

Factor the polynomial: x^2 - 5x + 6

• (x + 2)(x + 3)

• (x - 2)(x - 3)

• (x - 4)(x - 2)

• (x - 1)(x - 6)

Answer explanation

To factor x^2 - 5x + 6, we look for two numbers that multiply to 6 and add to -5. The numbers -2 and -3 fit this, so the polynomial factors to (x - 2)(x - 3). Thus, the correct answer is (x - 2)(x - 3).

Simplify the fraction: (2x^2 + 4x) / (2x)

• x^2 + 2x

• x - 2

• x + 2

• 2x + 4

Answer explanation

To simplify (2x^2 + 4x) / (2x), factor out 2x from the numerator: 2x(x + 2). Then, cancel 2x in the numerator and denominator, resulting in x + 2. Thus, the correct answer is x + 2.

Factor the polynomial: 3x^2 + 12x

• 3(x^2 + 4x)

• x(3x + 12)

• (3x + 12)(x)

• 3x(x + 4)

Answer explanation

To factor 3x^2 + 12x, first factor out the greatest common factor, which is 3x. This gives us 3x(x + 4). The correct choice is 3x(x + 4).

Simplify the fraction: (x^2 - 9) / (x + 3)

• x^2 + 3

• x + 3

• x - 3

• x - 9

Answer explanation

To simplify (x^2 - 9) / (x + 3), factor the numerator: (x - 3)(x + 3). Cancel (x + 3) from numerator and denominator, leaving x - 3. Thus, the simplified fraction is x - 3.

Factor the polynomial: x^2 + 7x + 10

• (x + 1)(x + 10)

• (x - 2)(x - 5)

• (x + 3)(x + 4)

• (x + 2)(x + 5)

Answer explanation

To factor x^2 + 7x + 10, we look for two numbers that multiply to 10 and add to 7. The numbers 2 and 5 fit this, so the polynomial factors to (x + 2)(x + 5).

Simplify the fraction: (4x^2 - 16) / (2x)

• (4x - 16) / (2x)

• (4x^2 + 16) / (2x)

• (2x^2 - 8) / (x)

• 2(x - 2)(x + 2) / x

Answer explanation

To simplify (4x^2 - 16) / (2x), factor the numerator: 4(x^2 - 4) = 4(x - 2)(x + 2). Thus, (4(x - 2)(x + 2)) / (2x) simplifies to 2(x - 2)(x + 2) / x, which is the correct answer.

Factor the polynomial: 2x^2 - 8x

• 2x(x + 4)

• x(2x - 8)

• 2x(x - 4)

• 2(x^2 - 4x)

Answer explanation

To factor 2x^2 - 8x, first factor out the common term 2x. This gives 2x(x - 4). The correct choice is 2x(x - 4), as it simplifies the polynomial correctly.

Simplify the fraction: (x^2 - 1) / (x - 1)

• x - 1

• x + 1

• x^2 + 1

• 1/(x - 1)

Answer explanation

To simplify (x^2 - 1) / (x - 1), factor the numerator: (x - 1)(x + 1). Cancel (x - 1) from numerator and denominator, leaving x + 1. Thus, the simplified fraction is x + 1.

Factor the polynomial: x^2 - 4x + 4

• (x + 2)(x - 2)

• (x - 2)(x + 2)

• (x - 4)(x + 1)

• (x - 2)^2

Answer explanation

The polynomial x^2 - 4x + 4 can be factored as (x - 2)(x - 2), which is (x - 2)^2. This shows it is a perfect square trinomial, confirming that the correct answer is (x - 2)^2.

Simplify the fraction: (3x^2 + 6x) / (3x)

• x^2 + 2x

• x - 2

• x + 2

• 3x + 6

Answer explanation

To simplify (3x^2 + 6x) / (3x), factor out 3x from the numerator: 3x(x + 2). Then, cancel 3x in the numerator and denominator, resulting in x + 2. Thus, the correct answer is x + 2.