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2. 5.4 Multiplying Polynomials

3. Multiply. a. (3x 2 )( – 2x) = (3)( – 2)(x 2 · x)= – 6x 3 b. 4x(4x 3 + 8) c. Example

4. Multiply.

5. To multiply any two polynomials, use the distributive property and multiply each term of one polynomial by each term of the other polynomial. Then combine any like terms. Multiplying Two Polynomials

6. Multiply. a. b. (2x – 4)(7x + 5)= 2x(7x + 5) – 4(7x + 5) = 14x 2 + 10x – 28x – 20 = 14x 2 – 18x – 20 Example

7. Multiply (a + 2)(a 3 – 3a 2 + 7). (a + 2)(a 3 – 3a 2 + 7) = a(a 3 – 3a 2 + 7) + 2(a 3 – 3a 2 + 7) = a 4 – 3a 3 + 7a + 2a 3 – 6a 2 + 14 = a 4 – a 3 – 6a 2 + 7a + 14 Example

8. Multiply (2y 2 + 5)(y 2 + 3y + 4) vertically. Example

9. The FOIL Method When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method. F – product of First terms O – product of Outside terms I – product of Inside terms L – product of Last terms

10. = y 2 – 8y – 48 Multiply (y – 12)(y + 4). (y – 12)(y + 4) Product of First terms is y 2 Product of Outside terms is 4y Product of Inside terms is – 12y Product of Last terms is – 48 (y – 12)(y + 4) = y 2 + 4y – 12y – 48 F O I L Example

11. Multiply (2x – 4)(7x + 5) (2x – 4)(7x + 5) = = 14x 2 + 10x – 28x – 20 F 2x(7x) F + 2x(5) O – 4(7x) I – 4(5) L O I L = 14x 2 – 18x – 20 Example

12. Multiply (3x – 7y)(7x + 2y) (3x – 7y)(7x + 2y)= (3x)(7x + 2y) – 7y(7x + 2y) = 21x 2 + 6xy – 49xy + 14y 2 = 21x 2 – 43xy + 14y 2 Example

13. A binomial squared is equal to the square of the first term plus or minus twice the product of both terms plus the square of the second term. (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2 Squaring a Binomial

14. Example Multiply. (x + 6) 2 F OI L (x + 6) 2 = x 2 + 6x + 6x + 36 = x 2 + 12x + 36 The inner and outer products are the same. = (x + 6)(x + 6)

15. Example a. (12a – 3) 2 = 144a 2 – 72a + 9 = (12a) 2 – 2(12a)(3) + (3) 2 b. (x + y) 2 = x 2 + 2xy + y 2 Multiply.

16. Product of the Sum and Difference of Two Terms The product of the sum and difference of two terms is the square of the first term minus the square of the second term. (a + b)(a – b) = a 2 – b 2 Multiplying the Sum and Difference of Two Terms

17. Example Multiply. = (2x)(2x) + (2x)(– 4) + (4)(2x) + (4)(– 4) F OI L (2x + 4)(2x – 4) = 4x 2 + (– 8x) + 8x + (–16) = 4x 2 – 16 The inner and outer products cancel.

18. Example Multiply. a. (5a + 3)(5a – 3) = 25a 2 – 9 = (5a) 2 – 3 2 b. (8c + 2d)(8c – 2d) = 64c 2 – 4d 2 = (8c) 2 – (2d) 2

19. Example Multiply

20. Example Multiply

21. Techniques of multiplying polynomials are often useful when evaluating polynomial functions at polynomial values. If f(x) = 2x 2 + 3x – 4, find f(a + 3). We replace the variable x with a + 3 in the polynomial function. f(a + 3) = 2(a + 3) 2 + 3(a + 3) – 4 = 2(a 2 + 6a + 9) + 3a + 9 – 4 = 2a 2 + 12a + 18 + 3a + 9 – 4 = 2a 2 + 15a + 23 Evaluating Polynomials Example