6.4: Multiply polynomial expressions
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We can multiply polynomials of different forms, but the method will be the same. We first look at multiplying monomials, multiplying a monomial and polynomial, and then finish with multiplying polynomials. We are using the product rule of exponents and the distributive property.
Multiply a Polynomial Expression by a Monomial
Multiply: (4x3y4z)(2x2y6z3)
Solution
(4x3y4z)(2x2y6z3)Rewrite without parenthesis4x3y4z⋅2x2y6z3Multiply coefficients and apply the product rule of exponents4⋅2⋅x3+2y4+6z1+3Simplify8x5y10z4Product
Multiply: 4x3(5x2−2x+5)
Solution
4x3(5x2−2x+5)Distribute 4x34x3⋅5x2−4x3⋅2x+4x3⋅5Multiply and apply the product rule of exponents20x5−8x4+20x3Product
Multiply: 2a3b(3ab2−4a)
Solution
2a3b(3ab2−4a)Distribute 2a3b2a3b⋅3ab2−2a3b⋅4aMultiply and apply the product rule of exponents6a3+1b1+2−8a3+1bSimplify6a4b3−8a4bProduct
Multiplying with Binomials
There are several different methods for multiplying polynomials, all of which result in the same answer. We discuss multiplying by distribution and the FOIL method.
Multiply: (3x+5)(x+13)
Solution
We will multiply using distribution and then simplify.
(3x+5)(x+13)Distribute 3x and 5 to (x+13)3x(x+13)+5(x+13)Distribute3x2+39x+5x+65Combine like terms3x2+44x+65Product
Multiply: (4x+7y)(3x−2y)
Solution
We will multiply using the FOIL method. FOIL is an acronym and represents
First-Multiply the first terms in each parenthesisOuter-Multiply the outer terms in each parenthesisInner-Multiply the inner terms in each parenthesisLast-Multiply the last terms in each parenthesis
=12x2−8xy+21xy−14y2=12x2+13xy+14y2
Multiplying with Trinomials
A trinomial is a polynomial with three terms. Usually, in Algebra, a trinomial takes the form of ax2+bx+c, where a,b, and c are coefficients.
Multiply: (2x−5)(4x2−7x+3)
Solution
Since we are multiplying a binomial with a trinomial, we can use distribution to multiply.
(2x−5)(4x2−7x+3)Distribute 2x and −5 to (4x2−7x+3)2x⋅(4x2−7x+3)−5(4x2−7x+3)Distribute8x3−14x2+6x−20x2+35x−15Combine like terms8x3−34x2+41x−15Product
Multiply: (5x2+x−10)(3x2−10x−6)
Solution
Since we are multiplying a trinomial with a trinomial, then we can use distribution to multiply.
(5x2+x−10)(3x2−10x−6)Distribute 5x2,x, and −10to (3x2−10x−6)5x2⋅(3x2−10x−6)+x(3x2−10x−6)−10(3x2−10x−6)Distribute15x4−50x3−30x2+3x3−10x2−6x−30x2+100x+60Combine like terms15x4−47x3−70x2+94x+60Product
Multiplying Monomials and Binomials
Multiply: 3x(2x−4)(x+5)
Solution
We first use FOIL to multiply the binomials and then distribute the 3x.
=3x(2x2+10x−4x−20)=3x(2x2+6x−20)
Lastly, we distribute 3x:
3x(2x2+6x−20)3x⋅2x2+3x⋅6x−3x⋅206x3+18x2−60x
Thus, the product is 6x3+18x2−60x.
In Example 6.4.8 , a common error is distributing the 3x first and into both parenthesis. While we can distribute the 3x into the (2x−4) factor, we cannot distribute into both factors. We recommend to multiply the binomials first, then distribute the monomial last.
Multiplying Polynomial Functions
We can multiply two polynomial functions the same way we multiply polynomial expressions, except, now, we have functions. The method is the same, but the notation and problems change.
If f and g are two functions of x, then (f⋅g)(x)=f(x)⋅g(x)
Let f(x)=2x−1 and g(x)=x+4. Find (f⋅g)(x).
Solution
We start by applying the definition, then simplify completely.
(f⋅g)(x)=f(x)⋅g(x)Apply the definition(f⋅g)(x)=(2x−1)⋅(x+4)Multiply two binomials(f⋅g)(x)=2x2+8x−x−4Combine like terms(f⋅g)(x)=2x2+7x−4The product of f and g
Multiply Polynomial Expressions Homework
Multiply and simplify.
6(p−7)
2(6x+3)
5m4(4m+4)
(4n+6)(8n+8)
(8b+3)(7b−5)
(4x+5)(2x+3)
(3v−4)(5v−2)
(6x−7)(4x+1)
(5x+y)(6x−4y)
(x+3y)(3x+4y)
(7x+5y)(8x+3y)
(r−7)(6r2−r+5)
(6n−4)(2n2−2n+5)
(6x+3y)(6x2−7xy+4y2)
(8n2+4n+6)(6n2−5n+6)
(5k2+3k+3)(3k2+3k+6)
3(3x−4)(2x+1)
3(2x+1)(4x−5)
7(x−5)(x−2)
6(4x−1)(4x+1)
4k(8k+4)
3n2(6n+7)
3(4r−7)
(2x+1)(x−4)
(r+8)(4r+8)
(7n−6)(n+7)
(6a+4)(a−8)
(5x−6)(4x−1)
(2u+3v)(8u−7v)
(8u+6v)(5u−8v)
(5a+8b)(a−3b)
(4x+8)(4x2+3x+5)
(2b−3)(4b2+4b+4)
(3m−2n)(7m2+6mn+4n2)
(2a2+6a+3)(7a2−6a+1)
(7u2+8uv−6v2)(6u2+4uv+3v2)
5(x−4)(2x−3)
2x(4x+1)(2x−6)
5x(2x−1)(4x+1)
3x2(2x+3)(6x+9)
Perform the indicated operations given the set of functions.
Let g(x)=4x+5 and h(x)=x2+5x, find (g⋅h)(x).
Let p(t)=t−4 and r(t)=2t, find (p⋅r)(t).
Let f(n)=−2n2−5n and k(n)=n+5, find (f⋅k)(n).