pantropiko brrrrt

1. MATHEMATICS 8
LESSON 1

2. Multiplication of
Polynomials
Recall

3. Algebraic expressions that are to be multiplied in a
multiplication sentence are called factors. The
resulting expression is called the product.
Consider the examples below.
Factors Product
1. 2𝑥 𝑎 + 𝑏 + 𝑐 = 2𝑎𝑥 + 2𝑏𝑥 + 2𝑐𝑥
2. 𝑥 + 4 𝑥 + 3 = 𝑥2
+ 7𝑥 + 12
3. 𝑥 + 5 𝑥2
− 2𝑥 + 3 = 𝑥3
+ 3𝑥2
− 7𝑥 + 15

4. To find the product of two monomials, some
rules on exponents need to be applied.
Such rules are …
𝒂𝒎
× 𝒂𝒏
= 𝒂𝒎+𝒏
(𝒂𝒎
)𝒏
= 𝒂𝒎𝒏
(𝒂𝒃)𝒏
= 𝒂𝒏
𝒃𝒏
where m and n are both positive integers.

5. Examples
1. 2𝑥 𝑎 + 𝑏 + 𝑐
= 2𝑥 ∙ 𝑎 + 2𝑥 ∙ 𝑏 + 2𝑥 ∙ 𝑐 Distributive Property
= 2𝑎𝑥 + 2𝑏𝑥 + 2𝑐𝑥
2. x + 4 x + 3
= 𝑥 + 4 𝑥 + 𝑥 + 4 3
= 𝑥 ∙ 𝑥 + 4 ∙ 𝑥 + 𝑥 ∙ 3 + 4 ∙ 3
= 𝑥2
+ 4𝑥 + 3𝑥 + 12 Combine similar terms
= 𝑥2
+ 7𝑥 + 12 Simplify.

6. Special Products
There are some special products
that come in special forms.

7. Perfect Square Trinomial
The square of a binomial, called perfect
square trinomial, is the square of the first
term, plus or minus twice the product of the
two terms, plus the square of the second
term. In symbols,
(𝑎 + 𝑏)2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
(𝑎 − 𝑏)2
= 𝑎2
− 2𝑎𝑏 + 𝑏2

8. Examples
(𝑥 + 4)2
= 𝑥 + 4 𝑥 + 4
= 𝑥2
+ 4𝑥 + 4𝑥 + 16
= 𝑥2
+ 2 4𝑥 + 16
= 𝑥2
+ 8𝑥 + 16
(3𝑚 − 4)2
= 3𝑚 − 4 3𝑚 − 4
= 9𝑚2
− 12𝑚 − 12𝑚 + 16
= 9𝑚2
− 2 12𝑚 + 16
= 9𝑚2
− 24𝑚 + 16

9. Difference of Two Squares
The product of the sum and difference
of two quantities is a difference of two
squares, that is, the square of the first
quantity minus the square of the
second. In symbols,
𝑎 + 𝑏 𝑎 − 𝑏 = 𝑎2
− 𝑏2

10. Examples
𝑥 + 2 𝑥 − 2 = 𝑥2
+ 2𝑥 − 2𝑥 − 4
= 𝑥2
− 4
2𝑥 − 3 2𝑥 + 3 = 4𝑥2
− 6𝑥 + 6𝑥 − 9
= 4𝑥2
− 9
1 + 𝑥 1 − 𝑥 = 1 − 𝑥 + 𝑥 − 𝑥2
= 1 − 𝑥2

11. Sum and Difference of Two Cubes
(𝑎 + 𝑏)(𝑎2−𝑎𝑏+𝑏2) = 𝑎3 + 𝑏3
(𝑎 − 𝑏)(𝑎2+𝑎𝑏+𝑏2) = 𝑎3 − 𝑏3

12. Examples
(𝑎 + 𝑏)(𝑎2
−𝑎𝑏+𝑏2
)
= 𝑎(𝑎2
− 𝑎𝑏 + 𝑏2
) + 𝑏(𝑎2
− 𝑎𝑏 + 𝑏2
)
= 𝑎3
− 𝑎2
𝑏 +𝑎𝑏2
+𝑎2
𝑏 − 𝑎𝑏2
+𝑏3
= 𝑎3
+ 𝑏3
(𝑎 − 𝑏)(𝑎2
+𝑎𝑏+𝑏2
)
= 𝑎 𝑎2
+ 𝑎𝑏 + 𝑏2
− 𝑏(𝑎2
+ 𝑎𝑏 + 𝑏2
)
= 𝑎3
+ 𝑎2
𝑏 +𝑎𝑏2
−𝑎2
𝑏 − 𝑎𝑏2
−𝑏3
= 𝑎3
− 𝑏3

13. PRACTICE!
LET’S GO! 

14. Find the product in each of the following.
1. (𝑥 − 4)2
2. (3 + 7𝑥)(3 − 7𝑥)

15. ANSWERS
1. (𝑥 − 4)2
= 𝑥 − 4 𝑥 − 4
= 𝑥2 − 4𝑥 − 4𝑥 + 16
= 𝑥2 + 2 −4𝑥 + 16
= 𝑥2 − 8𝑥 + 16

16. ANSWERS
2. (3 + 7𝑥)(3 − 7𝑥)
= 9 − 21𝑥 + 21𝑥 − 49𝑥2
= 9 − 49𝑥2