Multiplication and division

1. MULTIPLYING AND DIVIDING
POLYNOMIALS

2. MULTIPLICATION OF POLYNOMIALS
• If all the polynomials are monomials, use the associative and
commutative property.
• If any of the polynomials are not monomials, use the distributive
property before the associative and commutative properties. Then
combine like terms.

3. Example: Multiplying Polynomials
1. (3𝑥2)(-2x) = (3)(-2)(𝑥2  x) = -6 𝑥3
2. (4𝑥2)(3𝑥2 - 2x + 5)
= (4𝑥2
)(3𝑥2
) - (4𝑥2
)(2x) + (4𝑥2
)(5) Distributive Property
= 12𝑥4 - 8𝑥3 + 20𝑥2 Multiply the
monomials

4. Example: Multiplying Polynomials
3. (2x – 4)(7x + 5)
= 2x(7x + 5) – 4(7x + 5)
= 14 𝑥2 + 10x – 28x - 20
= 14 𝑥2 - 18x - 20

5. Example: Multiplying Polynomials
4. 3𝑥 + 4 2
= (3x + 4)(3x + 4)
= (3x)(3x + 4) + 4(3x + 4)
= 9 𝑥2 + 12x + 12x + 16
= 9 𝑥2 + 24x + 16

6. Example: Multiplying Polynomials
5. (a + 2)(𝑎3
- 3 𝑎2
+ 7)
= a(𝑎3 - 3 𝑎2 + 7) + 2(𝑎3 - 3 𝑎2 + 7)
= 𝑎4 - 3 𝑎3 + 7a + 2 𝑎3 - 6 𝑎2 + 14
= 𝑎4 - 𝑎3 - 6 𝑎2 + 7a + 14

7. Example: Multiplying Polynomials
6. 5𝑥 − 2𝑧 2
= (5x – 2z)(5x – 2z)
= (5x)(5x – 2z) – (2z)(5x – 2z)
= 25 𝑥2 - 10xz – 10xz + 4 𝑧2
= 25 𝑥2 - 20xz + 4 𝑧2

8. Example: Multiplying Polynomials
7. (2 𝑥2
+ x – 1)(𝑥2
+ 3x + 4)
= (2 𝑥2)(𝑥2 + 3x + 4) + (x)(𝑥2 + 3x + 4) – 1(𝑥2 + 3x + 4)
= 2 𝑥4 + 6 𝑥3 + 8 𝑥2 + 𝑥3 + 3 𝑥2 + 4x - 𝑥2 - 3x – 4
= 2 𝑥4 + 7 𝑥3 + 10 𝑥2 + x - 4

9. SPECIAL PRODUCTS
When multiplying 2 binomials, the distributive property can be easily
remembered as the FOIL method.
F – product of FIRST term
O – product of OUTSIDE term
I – product of INSIDE term
L – product of LAST term

10. Example: Special Products
1. (y – 12)(y + 4)
(y – 12)(y + 4) Product of first terms is 𝑦2
.
(y – 12)(y + 4) Product of outside terms is 4y.
(y – 12)(y + 4) Product of inside terms is -12y.
(y – 12)(y + 4) Product of last terms is -48.
(y – 12)(y + 4) = 𝑦2
+ 4y – 12y – 48
= 𝑦2 - 8y - 48

11. Example: Special Products
2. (2x – 4)(7x + 5)
= 2x(7x) + 2x(5) – 4(7x) – 4(5)
= 14𝑥2 + 10x – 28x – 20
= 14𝑥2 - 18x – 20

12. SPECIAL PRODUCTS
In the process of using the FOIL method on products of certain types of
binomials, we see specific patterns that lead to special products.
SQUARING A BINOMIAL
𝒂 + 𝒃 𝟐
= 𝒂𝟐
+ 𝟐𝒂𝒃 + 𝒃𝟐
𝒂 − 𝒃 𝟐 = 𝒂𝟐 − 𝟐𝒂𝒃 + 𝒃𝟐
MULTIPLYING THE SUM AND DIFFERENCE OF TWO TERMS
(a + b)(a – b) = 𝒂𝟐 - 𝒃𝟐

13. Example: Special Products
1. 𝑥 + 2 2
= (x + 2)(x + 2)
= 𝑥2 + 2x + 2x + 4
= 𝑥2 + 4x + 4
𝒂 + 𝒃 𝟐 = 𝒂𝟐 + 𝟐𝒂𝒃 + 𝒃𝟐
𝑥 + 2 2 = (𝑥)2 + 2 (x)(2) + (2)2
= 𝑥2 + 4x + 4

14. Example: Special Products
1. (x + 2)(x – 2) = 𝑥2
+ 2x – 2x – 4
= 𝑥2 – 4
(a + b)(a – b) = 𝒂𝟐 - 𝒃𝟐
(x + 2)(x – 2) = (𝑥)2 - (2)2
= 𝒙𝟐 - 4

15. Example: Special Products
1. 𝑥 − 2 2 =
2. 𝑥 + 3 2 =
3. (x + 4)(x – 4) =

16. DIVIDING POLYNOMIALS
When dividing a polynomial by a monomial, divide each term of the
polynomial separately by the monomial.
Example:
−12𝑎3+36𝑎 −15
3𝑎
=
−12𝑎3
3𝑎
+
36𝑎
3𝑎
-
15
3𝑎
= -4𝑎2 + 12 -
5
𝑎

17. DIVIDING POLYNOMIALS
- Dividing a polynomial by a polynomial other than a
monomial uses a “long division” technique that is like the
process known as long division in dividing two numbers.

21. DIVIDING POLYNOMIALS
One method of dividing polynomials is the SYNTHETIC DIVISION.
Synthetic division is generally used, however, not for dividing out factors but for
finding zeroes (or roots) of polynomials.

22. DIVIDING POLYNOMIALS
Advantages and Disadvantages of Synthetic Division Method
The advantages of using the synthetic division method are:
• It requires only a few calculation steps
• The calculation can be performed without variables
• Unlike the polynomial long division method, this method is a less error-prone
method
The only disadvantage of the synthetic division method is that this method is only
applicable if the divisor of the polynomial expression is a linear factor.

23. Using the long division:
𝒙𝟐+ 5x + 6  x – 1 =

24. Using the synthetic division:
First, take the polynomial, and write the coefficients ONLY inside in an upside–
down division–type symbol.
𝒙𝟐+ 5x + 6  x – 1 =

25. Using the synthetic division:
Put the test zero, in our case x = 1, at the left, next to the (top) row of numbers:
𝒙𝟐+ 5x + 6  x – 1 =

26. Using the synthetic division:
Take the first number that's on the inside, the number that represents the polynomial's
leading coefficient, and carry it down, unchanged, to below the division symbol:
𝒙𝟐+ 5x + 6  x – 1 =

27. Using the synthetic division:
Multiply this carry-down value by the test zero on the left, and carry the result
up into the next column inside:
𝒙𝟐+ 5x + 6  x – 1 =

28. Using the synthetic division:
Add down the column:
𝒙𝟐+ 5x + 6  x – 1 =

29. Using the synthetic division:
Multiply the previous carry-down value by the test zero, and carry the new
result up into the last column:
𝒙𝟐+ 5x + 6  x – 1 =

30. Using the synthetic division:
Add down the column:
This last carry-down value is the remainder.
𝒙𝟐+ 5x + 6  x – 1 =

31. You try!
1. 𝑥2
+ 5x + 6  x – 1
2.
2𝑥3 −5𝑥2+3𝑥+7
𝑥 −2
3.
2𝑥3+ 5𝑥2+9
𝑥+3

32. DIVIDING POLYNOMIALS
DIVISION OF A MONOMIAL BY ANOTHER MONOMIAL
40𝑥2
10𝑥
=
2 ∙ 2 ∙ 2 ∙ 5 ∙ 𝑥 ∙ 𝑥
2 ∙ 5 ∙ 𝑥
= 2  2  x
= 4x

33. DIVIDING POLYNOMIALS
DIVISION OF A POLYNOMIAL BY MONOMIAL
24x3 – 12xy + 9x  3x
=
24x3
3𝑥
−
12𝑥𝑦
3𝑥
+
9𝑥
3𝑥
= 8𝑥2 - 4y + 3

34. DIVIDING POLYNOMIALS
DIVISION OF A POLYNOMIAL BY BINOMIAL
Divide: 3x3 – 8x + 5 by x – 1

35. DIVIDING POLYNOMIALS
DIVISION OF A POLYNOMIAL
BY ANOTHER POLYNOMIAL
Divide: x2 + 2x + 3x3 + 5 by 1 + 2x + x2