Factoring polynomials

High School Math 1 · Unit 8 · factoring-polynomials · Teacher edition

factoring-polynomials

Name ______________________________Date ______________

1.
Which is the complete factorization of $12x^{3} - 18x^{2}$ ?
1. (A)
  $6x(2x^{2} - 3x)$
2. (B)
  $3x^{2}(4x - 6)$
3. (C)
  $2x^{2}(6x - 9)$
4. (D)
  $6x^{2}(2x - 3)$
2.
Factor $6x^{2} - 5x - 6$ completely.
Answer: ______________
3.
A rectangle has area $x^{2} + 7x + 12$ square units and width $x + 3$ units. Write an expression for its length.
Answer: ______________
4.
Factor $x^{2} - 9$ .
1. (A)
  $(x - 3)^{2}$
2. (B)
  $(x + 3)^{2}$
3. (C)
  $(x - 3)(x + 3)$
4. (D)
  $(x - 9)(x + 1)$
5.
The trinomial $x^{2} + bx + 48$ can be factored as $(x + p)(x + q)$ , where $p$ and $q$ are positive integers. How many different values of $b$ are possible?
1. (A)
  $4$
2. (B)
  $5$
3. (C)
  $6$
4. (D)
  $8$
5. (E)
  $10$

Answer key — Factoring polynomials

1.
(D)
$6x^{2}(2x - 3)$
factoring-polynomials-01
GCF $= 6x^2$ , so $12x^3 - 18x^2 = \mathbf{6x^{2}(2x - 3)}$ . (The other three are all true products, but each leaves a common factor inside the bracket — $x$ , $2$ , or $3$ — so they are not complete.)
2.
$(2x-3)(3x+2)$ factoring-polynomials-02
$6x^2 - 5x - 6 = 6x^2 + 4x - 9x - 6 = 2x(3x + 2) - 3(3x + 2) = \mathbf{(2x - 3)(3x + 2)}$ . Check the middle term: $2x \cdot 2 + (-3)(3x) = 4x - 9x = -5x$ . ✓
3.
$x+4$ factoring-polynomials-03
$x^2 + 7x + 12 = (x + 3)(x + 4)$ . The width is $x + 3$ , so the length is $\mathbf{x + 4}$ .
4.
(C)
$(x - 3)(x + 3)$
factoring-polynomials-04
$x^2 - 9 = (x - 3)(x + 3)$ ; the middle terms $3x$ and $-3x$ cancel. Answer: $\mathbf{(x - 3)(x + 3)}$ . ( $(x-3)^2 = x^2 - 6x + 9$ and $(x+3)^2 = x^2 + 6x + 9$ both have a middle term; $(x - 9)(x + 1) = x^2 - 8x - 9$ .)
5.
(B)
$5$
factoring-polynomials-05
The sums are $49, 26, 19, 16, 14$ — all different — so there are $\mathbf{5}$ possible values of $b$ . ( $10$ counts each unordered pair twice; $4$ misses a pair.)