Factoring patterns Practice Questions (Grade 9)

Factoring patterns. Game on.

Factoring patterns is a grade 9 math skill aligned to Common Core standard HSA.SSE.A.2: use the structure of an expression to identify ways to rewrite it. Below are 8 practice questions with answers and step-by-step explanations, drawn from the 15 factoring patterns problems our math games drill.

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Warm-upeasy

$x^3 - 8 = 0$ . What is the real root?

$x^3 - 8 = (x - 2)(x^2 + 2x + 4) = 0$ . The quadratic factor has no real roots, so the only real root is $x = 2$ .

Mid-gameeasy

$x^2 - 49 = 0$ . What is the positive root?

$x^2 - 49 = (x - 7)(x + 7) = 0$ , so $x = 7$ or $x = -7$ . The positive root is $7$ .

Mid-gameeasy

$x^{2} + 10x + 25 = (x + a)^{2}$ with $a > 0$ . What is $a$ ?

Half of $10$ is $5$ and $5^{2} = 25$ , so $x^{2} + 10x + 25 = (x + 5)^{2}$ and $a = 5$ .

Mid-gameeasy

$x^{3} + 27 = (x + 3)(x^{2} - 3x + c)$ . What is $c$ ?

Sum of cubes: $x^{3} + 27 = (x + 3)(x^{2} - 3x + 9)$ , so $c = 3^{2} = 9$ .

Mid-gameeasy

$x^3 - 27 = 0$ . What is the real root?

$x^3 - 27 = (x - 3)(x^2 + 3x + 9) = 0$ . The quadratic factor has no real roots, so the only real root is $x = 3$ .

Mid-gameeasy

$x^2 - 81 = 0$ . What is the positive root?

$x^2 - 81 = (x - 9)(x + 9) = 0$ , so $x = 9$ or $x = -9$ . The positive root is $9$ .

Mid-gameeasy

If $x^{2} - 14x + 49 = (x - a)^{2}$ with $a > 0$ , find $a$ .

Half of $14$ is $7$ and $7^{2} = 49$ , so $x^{2} - 14x + 49 = (x - 7)^{2}$ and $a = 7$ .

Buzzer beatereasy

In the factorization $x^{3} - 125 = (x - 5)(x^{2} + 5x + c)$ , what is $c$ ?

Difference of cubes: $x^{3} - 125 = (x - 5)(x^{2} + 5x + 25)$ , so $c = 5^{2} = 25$ .

Factoring patterns. Game on.

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