What is the product of the polynomials below?
$\left(5 x^2+5 x+7\right)(8 x+6)$
A. $40 x^3+60 x^2+86 x+42$
B. $40 x^3+10 x^2+86 x-42$
C. $40 x^3+70 x^2+86 x+42$
D. $48 x^3+70 x^2+86 x+42$
Answer (1)
Multiply each term of the first polynomial by each term of the second polynomial.
Expand each term: 5 x 2 ( 8 x + 6 ) = 40 x 3 + 30 x 2 , 5 x ( 8 x + 6 ) = 40 x 2 + 30 x , 7 ( 8 x + 6 ) = 56 x + 42 .
Combine like terms: 40 x 3 + 30 x 2 + 40 x 2 + 30 x + 56 x + 42 = 40 x 3 + 70 x 2 + 86 x + 42 .
The product of the polynomials is 40 x 3 + 70 x 2 + 86 x + 42 .
Explanation
Understanding the Problem We are given two polynomials, ( 5 x 2 + 5 x + 7 ) and ( 8 x + 6 ) , and we need to find their product. This involves multiplying each term of the first polynomial by each term of the second polynomial and then combining like terms.
Solution Plan To find the product of the polynomials ( 5 x 2 + 5 x + 7 ) and ( 8 x + 6 ) , we use the distributive property. This means we multiply each term in the first polynomial by each term in the second polynomial.
Multiplying the Polynomials We multiply the two polynomials as follows:
( 5 x 2 + 5 x + 7 ) ( 8 x + 6 ) = 5 x 2 ( 8 x + 6 ) + 5 x ( 8 x + 6 ) + 7 ( 8 x + 6 )
Expanding Each Term Now, we expand each term:
5 x 2 ( 8 x + 6 ) = 40 x 3 + 30 x 2
5 x ( 8 x + 6 ) = 40 x 2 + 30 x
7 ( 8 x + 6 ) = 56 x + 42
Combining Like Terms Next, we combine the expanded terms:
40 x 3 + 30 x 2 + 40 x 2 + 30 x + 56 x + 42
Now, we combine like terms:
40 x 3 + ( 30 x 2 + 40 x 2 ) + ( 30 x + 56 x ) + 42 = 40 x 3 + 70 x 2 + 86 x + 42
Final Result The product of the polynomials is 40 x 3 + 70 x 2 + 86 x + 42 . Comparing this with the given options, we see that it matches option C.
Examples
Polynomial multiplication is a fundamental concept in algebra and has many real-world applications. For example, if you are designing a rectangular garden where the length is represented by the polynomial ( 8 x + 6 ) and the width is represented by the polynomial ( 5 x 2 + 5 x + 7 ) , the area of the garden can be found by multiplying these two polynomials. The resulting polynomial, 40 x 3 + 70 x 2 + 86 x + 42 , represents the area of the garden in terms of x . This allows you to calculate the area for different values of x , which could represent different design parameters or constraints.
