Polynomials By C. D.Toliver

Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4 –Trinomials have three terms, x 2 + 3x + 4

Review: Collecting Like Terms You simplify polynomial expressions by collecting like terms Like terms have the same variable and the same exponent. 2a, 5a, -7a are like terms 2a, 5b, 6c are not like terms a 3, a 2, a are not like terms Constants are also like terms 2, 0.5, ¼ are like terms

Review Collecting Like Terms Example 1. Simplify 3x 2 + 5x – 2x 2 + 3x + 7 1x 2 + 8x + 7

Review: Collecting Like Terms Example 2. Simplify (5a + 7b + 6) + (- 8a - 9b + 5) 5a + 7b a + - 9b a + 7b a - 9b a – 2b + 11

Review: Collecting Like Terms Example 4. Simplify (5a + 7b + 6) – (8a - 6b + 5) 5a + 7b a - -6b -+5 5a + 7b a + 6b a + 13b + 1

Review: Distributive Property We also learned that parenthesis mean to multiply. We use the distributive property to multiply polynomials The distributive property says: a(b+c) = a(b) + a(c)

Review: Distributive Property Example 1 Multiply 3(x+y) 3(x) + 3(y) 3x +3y

Review: Distributive Property Example 2. Multiply 4(2x – 3) 4(2x) + 4(-3) 8x -12

Review: Distribute and Collect For more complex expressions you may need to distribute and collect like terms. Distribute first Then collect

Review: Distribute and Collect Example 1. Distribute and Collect 6(x + 5) - 2(2x – 8) 6(x) +6(5) -2(2x) -2(-8)Distribute 6x + 30 – 4x + 16Collect 2x + 46

Review: Distribute and Collect Example 2. Distribute and Collect 3(2x - 4) + 7(x – 2) 3(2x) + 3(-4) +7(x) + 7(-2)Distribute 6x x - 14Collect 13x - 26

Review: Distribute and Collect Example 3. Distribute and Collect 5(y - 3) + 4(6 - 2y) 5(y) +5(-3) +4(6) +4(-2y)Distribute 5y – 8yCollect -3y + 9

Multiply Polynomials In the previous examples, we were multiplying polynomials by a monomial, e.g., 3 (x+2) 3 is a monomial x+2 is a polynomial What happens when you multiply two polynomials, e.g., (x + 4)(x+2)?

Multiply Polynomials We will look at three different methods to multiply polynomials You may prefer one method over another Today we will practice all three methods

Multiply Polynomials Distributive Method Example 1. Multiply (x + 4)(x + 2) x(x+2) + 4(x+2) Distribute x(x) + x(2) + 4(x) +4(2)Distribute x 2 + 2x + 4x + 8Collect x 2 + 6x + 8

Multiply Polynomials Vertical Method Example 1. Multiply (x + 4)(x + 2) Rewrite vertically X + 4 X + 2 2x + 8 Multiply x 2 + 4x Multiply x 2 + 6x + 8 Combine

Multiply Polynomials Box Method Example 1. Multiply (x + 4)(x + 2)= x 2 + 6x + 8 x2x2 2x 4x8 x +2 x +4

Multiply Polynomials Distributive Method Example 2. Multiply (x - 3)(x + 5) x(x+5) - 3(x+5) Distribute x(x) + x(5) - 3(x) -3(5)Distribute x 2 + 5x - 3x - 15Collect x 2 + 2x - 15

Multiply Polynomials Vertical Method Example 2. Multiply (x - 3)(x + 5) Rewrite vertically X - 3 X + 5 5x - 15 Multiply x 2 - 3x Multiply x 2 + 2x - 15 Combine

Multiply Polynomials Box Method Example 2. Multiply (x - 3)(x + 5)= x 2 + 2x - 15 x2x2 -3x 5x-15 x -3 x +5

Multiply Polynomials Distributive Method Example 3. Multiply (2x + 1)(x - 4) 2x(x-4) + 1(x-4) Distribute 2x(x)+2x(-4)+1(x)+1(-4)Distribute 2x 2 - 8x + 1x -4Collect 2x 2 - 7x - 4

Multiply Polynomials Vertical Method Example 3. Multiply (2x + 1)(x - 4) Rewrite vertically 2x + 1 x x - 4 Multiply 2x 2 + 1x Multiply 2x 2 - 7x - 4 Combine

Multiply Polynomials Box Method Example 3. Multiply (2x + 1)(x - 4)= 2x 2 - 7x - 4 2x 2 1x -8x-4 2x +1 x -4

Multiply Polynomials Distributive Method Example 4. Multiply (2x + 3)(3x - 4) 2x(3x-4)+3(3x-4) Distribute 2x(3x)+2x(-4)+3(3x)+3(-4)Distribute 6x 2 - 8x + 9x - 12Collect 6x 2 + 1x - 12

Multiply Polynomials Vertical Method Example 4. Multiply (2x + 3)(3x - 4) Rewrite vertically 2X + 3 3X x - 12 Multiply 6x 2 + 9x Multiply 6x 2 + 1x - 12 Combine

Multiply Polynomials Box Method Example 4. Multiply (2x + 3)(3x - 4)= 6x 2 + 1x x 2 9x -8x-12 2x +3 3x -4