What is the product of the polynomials below?
[tex](8 x+8)(x+3)[/tex]
A. [tex]8 x^2+11 x+16[/tex]
B. [tex]8 x^2+32 x+24[/tex]
C. [tex]8 x^2+16 x+32[/tex]
D. [tex]8 x^2+34 x+64[/tex]
Answer (2)
Multiply the two polynomials using the distributive property: ( 8 x + 8 ) ( x + 3 ) = 8 x ( x ) + 8 x ( 3 ) + 8 ( x ) + 8 ( 3 ) .
Simplify the expression: 8 x 2 + 24 x + 8 x + 24 .
Combine like terms: 8 x 2 + 32 x + 24 .
The product of the polynomials is 8 x 2 + 32 x + 24 .
Explanation
Understanding the Problem We are given two polynomials, ( 8 x + 8 ) and ( x + 3 ) , and we need to find their product. This involves multiplying each term in the first polynomial by each term in the second polynomial.
Applying the Distributive Property To find the product of ( 8 x + 8 ) and ( x + 3 ) , we use the distributive property (often remembered as the FOIL method: First, Outer, Inner, Last). This means we multiply each term in the first polynomial by each term in the second polynomial:
( 8 x + 8 ) ( x + 3 ) = ( 8 x ) ( x ) + ( 8 x ) ( 3 ) + ( 8 ) ( x ) + ( 8 ) ( 3 )
Performing the Multiplication Now, we perform each of the multiplications:
( 8 x ) ( x ) = 8 x 2 ( 8 x ) ( 3 ) = 24 x ( 8 ) ( x ) = 8 x ( 8 ) ( 3 ) = 24
So, we have:
8 x 2 + 24 x + 8 x + 24
Combining Like Terms Next, we combine the like terms, which are the terms with the same variable and exponent. In this case, we combine 24 x and 8 x :
24 x + 8 x = 32 x
So, the expression becomes:
8 x 2 + 32 x + 24
Final Answer Therefore, the product of the polynomials ( 8 x + 8 ) and ( x + 3 ) is 8 x 2 + 32 x + 24 . Looking at the multiple-choice options, we see that this matches option B.
Examples
Polynomial multiplication is used in various real-world applications, such as calculating the area of a rectangular garden where the sides are expressed as polynomials. For example, if the length of a garden is ( x + 3 ) meters and the width is ( 8 x + 8 ) meters, the area of the garden can be found by multiplying these two polynomials, resulting in 8 x 2 + 32 x + 24 square meters. This calculation helps in determining the amount of fencing or fertilizer needed for the garden.
The product of the polynomials ( 8 x + 8 ) ( x + 3 ) is 8 x 2 + 32 x + 24 . This matches option B from the choices provided. Therefore, the answer is B.
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