Here are the step-by-step solutions for the problems on your worksheet. I will use the FOIL method (First, Outer, Inner, Last) to multiply the binomials. This means we multiply the terms in this order and then combine any "like" terms (terms with the same variable).
MULTIPLYING BINOMIALS
1. $(x - 4)(x + 10)$
*
First: $x \cdot x = x^2$
*
Outer: $x \cdot 10 = 10x$
*
Inner: $-4 \cdot x = -4x$
*
Last: $-4 \cdot 10 = -40$
*
Combine: $x^2 + 10x - 4x - 40$
*
Simplify: $10x - 4x = 6x$
2. $(x + 5)(x - 9)$
*
First: $x \cdot x = x^2$
*
Outer: $x \cdot -9 = -9x$
*
Inner: $5 \cdot x = 5x$
*
Last: $5 \cdot -9 = -45$
*
Combine: $x^2 - 9x + 5x - 45$
*
Simplify: $-9x + 5x = -4x$
3. $(2x - 4)(3x + 6)$
*
First: $2x \cdot 3x = 6x^2$
*
Outer: $2x \cdot 6 = 12x$
*
Inner: $-4 \cdot 3x = -12x$
*
Last: $-4 \cdot 6 = -24$
*
Combine: $6x^2 + 12x - 12x - 24$
*
Simplify: The middle terms ($12x$ and $-12x$) cancel each other out to become 0.
4. $(5x - 7y)(-x + 8y)$
*
First: $5x \cdot -x = -5x^2$
*
Outer: $5x \cdot 8y = 40xy$
*
Inner: $-7y \cdot -x = 7xy$ (Negative times negative is positive)
*
Last: $-7y \cdot 8y = -56y^2$
*
Combine: $-5x^2 + 40xy + 7xy - 56y^2$
*
Simplify: $40xy + 7xy = 47xy$
---
YOU TRY:
1. $(x - 7)(x - 1)$
*
First: $x \cdot x = x^2$
*
Outer: $x \cdot -1 = -1x$ (or just $-x$)
*
Inner: $-7 \cdot x = -7x$
*
Last: $-7 \cdot -1 = 7$
*
Combine: $x^2 - x - 7x + 7$
*
Simplify: $-x - 7x = -8x$
2. $(3x + 5)(2x - 8)$
*
First: $3x \cdot 2x = 6x^2$
*
Outer: $3x \cdot -8 = -24x$
*
Inner: $5 \cdot 2x = 10x$
*
Last: $5 \cdot -8 = -40$
*
Combine: $6x^2 - 24x + 10x - 40$
*
Simplify: $-24x + 10x = -14x$
3. $(x - 4y)^2$
* *Note: Squaring a binomial means multiplying it by itself: $(x - 4y)(x - 4y)$*
*
First: $x \cdot x = x^2$
*
Outer: $x \cdot -4y = -4xy$
*
Inner: $-4y \cdot x = -4xy$
*
Last: $-4y \cdot -4y = 16y^2$
*
Combine: $x^2 - 4xy - 4xy + 16y^2$
*
Simplify: $-4xy - 4xy = -8xy$
4. $(2x + 8y)(-7x - 3y)$
*
First: $2x \cdot -7x = -14x^2$
*
Outer: $2x \cdot -3y = -6xy$
*
Inner: $8y \cdot -7x = -56xy$
*
Last: $8y \cdot -3y = -24y^2$
*
Combine: $-14x^2 - 6xy - 56xy - 24y^2$
*
Simplify: $-6xy - 56xy = -62xy$
──────────────────────────────────────
Final Answer:
MULTIPLYING BINOMIALS
1. $x^2 + 6x - 40$
2. $x^2 - 4x - 45$
3. $6x^2 - 24$
4. $-5x^2 + 47xy - 56y^2$
YOU TRY:
1. $x^2 - 8x + 7$
2. $6x^2 - 14x - 40$
3. $x^2 - 8xy + 16y^2$
4. $-14x^2 - 62xy - 24y^2$
Parent Tip: Review the logic above to help your child master the concept of polynomial worksheets.
