Algebraic manipulation

Advanced Math 7 · Unit 4 · expand-factor-7 · Teacher edition

expand-factor-7

Name ______________________________Date ______________

1.
Simplify $\dfrac{1}{2}x + \dfrac{3}{4}x - 2x$ . Enter the coefficient of $x$ in the result.
Answer: ______________
2.
Expand and simplify $(2x - 3)(x + 5)$ .
Answer: ______________
3.
Factor $xy + 2x + 3y + 6$ completely.
Answer: ______________
4.
Expand $-2(x - 5)$ .
1. (A)
  $-2x - 10$
2. (B)
  $2x - 10$
3. (C)
  $-2x + 10$
4. (D)
  $-2x - 5$
5.
When $(2x + 3)(x - 4)$ is expanded and written as $ax^2 + bx + c$ , what is $a + b + c$ ?
1. (A)
  $-15$
2. (B)
  $-10$
3. (C)
  $-3$
4. (D)
  $1$
5. (E)
  $15$

Answer key — Algebraic manipulation

1.
-3/4expand-factor-7-01
$\frac{2}{4} + \frac{3}{4} - \frac{8}{4} = -\frac{3}{4}$ , so the expression is $-\frac{3}{4}x$ and the coefficient is $\mathbf{-\tfrac{3}{4}}$ .
2.
$2x^2+7x-15$ expand-factor-7-02
$(2x - 3)(x + 5) = 2x^2 + 10x - 3x - 15 = \mathbf{2x^2 + 7x - 15}$ . Check at $x = 1$ : $(-1)(6) = -6$ and $2 + 7 - 15 = -6$ . ✓
3.
$(x+3)(y+2)$ expand-factor-7-03
$xy + 2x + 3y + 6 = x(y + 2) + 3(y + 2) = \mathbf{(x + 3)(y + 2)}$ . Check by expanding: $xy + 2x + 3y + 6$ . ✓
4.
(C)
$-2x + 10$
expand-factor-7-04
$-2(x - 5) = -2x + 10$ : $\mathbf{-2x + 10}$ . ( $-2x - 10$ forgets that $-2 \times -5$ is positive; $2x - 10$ flips the wrong sign; $-2x - 5$ never multiplies the $5$ .)
5.
(A)
$-15$
expand-factor-7-05
$(2x + 3)(x - 4) = 2x^2 - 5x - 12$ , so $a + b + c = 2 - 5 - 12 = \mathbf{-15}$ . Shortcut: at $x = 1$ the product is $(5)(-3) = -15$ . ( $-10$ comes from $2x^2 - 12$ , the “first times first, last times last” error.)