University:
California State University, Northridge
Course:
MATH 320 | Foundations of Higher Mathematics
Academic year:

2024
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296

Pages:

15
Author:

Fradeaniwj4

§ 12.5 Multiplying Polynomials Multiplying Polynomials Multiplying polynomials • If all of the polynomials are monomials, use the associative and commutative properties. • If any of the polynomials are not monomials, use the distributive property before the associative and commutative properties. Then combine like terms. 40 Multiplying Polynomials Example Multiply each of the following. 1) (3x2)(– 2x) = (3)(– 2)(x2 · x) = – 6x3 2) (4x2)(3x2 – 2x + 5) = (4x2)(3x2) – (4x2)(2x) + (4x2)(5) = 12x4 – 8x3 + 20x2 (Distributive property) (Multiply the monomials) 3) (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5) = 14x2 + 10x – 28x – 20 = 14x2 – 18x – 20 41 Multiplying Polynomials Example Multiply (3x + 4)2 Remember that a2 = a · a, so (3x + 4)2 = (3x + 4)(3x + 4). (3x + 4)2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4) = 9x2 + 12x + 12x + 16 = 9x2 + 24x + 16 42 Multiplying Polynomials Example Multiply (a + 2)(a3 – 3a2 + 7). (a + 2)(a3 – 3a2 + 7) = a(a3 – 3a2 + 7) + 2(a3 – 3a2 + 7) a4 – 3a3 + 7a + 2a3 – 6a2 + 14 = a4 – a3 – 6a2 + 7a + 14 = 43 Multiplying Polynomials Example Multiply (3x – 7y)(7x + 2y) (3x – 7y)(7x + 2y) = (3x)(7x + 2y) – 7y(7x + 2y) = 21x2 + 6xy – 49xy + 14y2 = 21x2 – 43xy + 14y2 44 Multiplying Polynomials Example Multiply (5x – 2z)2 (5x – 2z)2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – 2z(5x – 2z) = 25x2 – 10xz – 10xz + 4z2 = 25x2 – 20xz + 4z2 45 Multiplying Polynomials Example Multiply (2x2 + x – 1)(x2 + 3x + 4) (2x2 + x – 1)(x2 + 3x + 4) = (2x2)(x2 + 3x + 4) + x(x2 + 3x + 4) – 1(x2 + 3x + 4) = 2x4 + 6x3 + 8x2 + x3 + 3x2 + 4x – x2 – 3x – 4 = 2x4 + 7x3 + 10x2 + x – 4 46 Multiplying Polynomials You can also use a vertical format in arranging the polynomials to be multiplied. In this case, as each term of one polynomial is multiplied by a term of the other polynomial, the partial products are aligned so that like terms are together. This can make it easier to find and combine like terms. 47 § 12.6 Special Products 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 49 Using the FOIL Method Example Multiply (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 F O I L (y – 12)(y + 4) = y2 + 4y – 12y – 48 = y2 – 8y – 48 50 Using the FOIL Method Example Multiply (2x – 4)(7x + 5) L F F O I L (2x – 4)(7x + 5) = 2x(7x) + 2x(5) – 4(7x) – 4(5) I O = 14x2 + 10x – 28x – 20 = 14x2 – 18x – 20 We multiplied these same two binomials together in the previous section, using a different technique, but arrived at the same product. 51 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 52 Special Products Although you will arrive at the same results for the special products by using the techniques of this section or last section, memorizing these products can save you some time in multiplying polynomials. 53