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# Polynomials

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1. Polynomials By C. D.Toliver

2. Polynomials • An algebraic expression with one or more terms • Monomials have one term, 3x • Binomials have two terms, 3x + 4 • Trinomials have three terms, x2 + 3x + 4

3. 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 a3, a2, a are not like terms • Constants are also like terms 2, 0.5, ¼ are like terms

4. ReviewCollecting Like Terms Example 1. Simplify 3x2 + 5x – 2x2 + 3x + 7 3x2+ 5x– 2x2+ 3x + 7 1x2+ 8x + 7

5. Review: Collecting Like Terms Example 2. Simplify (5a + 7b + 6) + (- 8a - 9b + 5) 5a+ 7b + 6 + - 8a + - 9b + + 5 5a+ 7b + 6 - 8a- 9b + 5 -3a– 2b + 11

6. Review: Collecting Like Terms Example 4. Simplify (5a + 7b + 6) – (8a - 6b + 5) 5a+ 7b + 6 - 8a- -6b -+5 5a+ 7b + 6 - 8a+ 6b - 5 -3a+ 13b + 1

7. 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)

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

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

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

11. 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+ 16 Collect 2x+ 46

12. 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- 12+ 7x- 14 Collect 13x- 26

13. 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- 15+24– 8y Collect -3y + 9

14. 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)?

15. 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

16. Multiply PolynomialsDistributive Method Example 1. Multiply (x + 4)(x + 2) x(x+2) + 4(x+2) Distribute x(x) + x(2) + 4(x) +4(2) Distribute x2+ 2x + 4x + 8 Collect x2+ 6x + 8

17. Multiply PolynomialsVertical Method Example 1. Multiply (x + 4)(x + 2) Rewrite vertically X + 4 X + 2 2x + 8 Multiply x2 + 4x Multiply x2 + 6x + 8 Combine

18. Multiply PolynomialsBox Method Example 1. Multiply (x + 4)(x + 2)= x2+ 6x + 8 x +2 x +4

19. Multiply PolynomialsDistributive Method Example 2. Multiply (x - 3)(x + 5) x(x+5) - 3(x+5) Distribute x(x) + x(5) - 3(x) -3(5) Distribute x2+ 5x - 3x - 15 Collect x2+ 2x - 15

20. Multiply PolynomialsVertical Method Example 2. Multiply (x - 3)(x + 5) Rewrite vertically X - 3 X + 5 5x - 15 Multiply x2 - 3x Multiply x2 + 2x - 15 Combine

21. Multiply PolynomialsBox Method Example 2. Multiply (x - 3)(x + 5)= x2+ 2x - 15 x -3 x +5

22. Multiply PolynomialsDistributive Method Example 3. Multiply (2x + 1)(x - 4) 2x(x-4) + 1(x-4) Distribute 2x(x)+2x(-4)+1(x)+1(-4) Distribute 2x2- 8x + 1x -4 Collect 2x2- 7x - 4

23. Multiply PolynomialsVertical Method Example 3. Multiply (2x + 1)(x - 4) Rewrite vertically 2x + 1 x - 4 -8x - 4 Multiply 2x2 + 1x Multiply 2x2 - 7x - 4 Combine

24. Multiply PolynomialsBox Method Example 3. Multiply (2x + 1)(x - 4)= 2x2- 7x - 4 2x +1 x -4

25. Multiply PolynomialsDistributive Method Example 4. Multiply (2x + 3)(3x - 4) 2x(3x-4)+3(3x-4) Distribute 2x(3x)+2x(-4)+3(3x)+3(-4)Distribute 6x2- 8x + 9x - 12 Collect 6x2+ 1x - 12

26. Multiply PolynomialsVertical Method Example 4. Multiply (2x + 3)(3x - 4) Rewrite vertically 2X + 3 3X - 4 -8x - 12 Multiply 6x2 + 9x Multiply 6x2 + 1x - 12 Combine

27. Multiply PolynomialsBox Method Example 4. Multiply (2x + 3)(3x - 4)= 6x2+ 1x - 12 2x +3 3x -4