| What to Do |
How to Do It |
| 1. The most basic property of algebra
necessary to multiplying polynomials is the distributive property. |
→
A(B + C) = AB + AC |
| 2. This can be extended to several terms inside
the parenthesis: (The three dots ... mean
and so on in the pattern.) |
→
A(B + C + D + ... )
= AB + AC + AD + ... |
| 3. Both multiplicands may be binomials, in which
case the property is usually referred to as the
“double distributive propertyâ€. |
→ (A + B)(C + D)
= A(C + D) + B(C + D)
= AC + AD + BC + BD |
| 4. Consider the following examples: |
|
| a) 2(x + 3) = |
→
2(x + 3)
= 2x + 2·3
= 2x + 6 |
| b) 3x(x-1) = |
→
3x(x - 1)
= 3x·x + 3x·(-1)
= 3x2 - 3x |
| c) - 5(x2 + 2x - 3) = |
→ -
5(x2 + 2x - 3)
= (- 5)(x2) + (- 5)(2x) + (- 5)(- 3)
= - 5x2 - 10x + 15 |
| d) (x + a)(y + b) = |
→ (x + a)(y + b)
= x(y + b) + a(y + b)
= xy + xb + ay + ab |
| e) (x + 2)(2x + 3) = With practice we can do the second
line in our head and go directly to the
third line. |
→
(x + 2) (2x + 3)
= x (2x + 3) + 2(2x + 3)
= x·2x + x·3
+ 2·2x + 2·3
= 2x2 + 3x + 4x + 6
= 2x2 + 7x + 6
|
| f) (2x + 3)(2x + 3) =
Later, skip this second line and →
go directly to this third line. → |
→
(2x + 3) (2x + 3)
= 2x·2x + 2x·3
+ 3·2x + 3·3
= 4x2 + 6x + 6x + 9
= 4x2 + 12x + 9 |
| g) (2x + 3)(2x - 3) =
Note the “equal and opposites†→
cancel out here. |
→ (2x + 3) (2x
- 3)
= 2x·2x + 2x·(-
3) + 3·2x + 3·(-
3)
= 4x2 - 6x + 6x - 9
= 4x2 - 9 |
