Multiply the polynomials.
$(x+2)\left(x^2-7 x+4\right)$
A. $x^3-7 x^2-10 x+8$
B. $x^3-5 x^2-10 x+8$
C. $x^3-7 x^2+4 x+8$
D. $x^3-5 x^2+4 x+8$
Answer (2)
Distribute each term of ( x + 2 ) to ( x 2 − 7 x + 4 ) .
Expand the expression: x ( x 2 − 7 x + 4 ) + 2 ( x 2 − 7 x + 4 ) = x 3 − 7 x 2 + 4 x + 2 x 2 − 14 x + 8 .
Combine like terms: x 3 − 5 x 2 − 10 x + 8 .
The correct answer is x 3 − 5 x 2 − 10 x + 8 .
Explanation
Understanding the Problem We are given two polynomials, ( x + 2 ) and ( x 2 − 7 x + 4 ) , 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.
Expanding the Polynomials To multiply the polynomials ( x + 2 ) and ( x 2 − 7 x + 4 ) , we distribute each term of the first polynomial to each term of the second polynomial: ( x + 2 ) ( x 2 − 7 x + 4 ) = x ( x 2 − 7 x + 4 ) + 2 ( x 2 − 7 x + 4 ) Now, we expand each term: x ( x 2 − 7 x + 4 ) = x 3 − 7 x 2 + 4 x 2 ( x 2 − 7 x + 4 ) = 2 x 2 − 14 x + 8
Combining Like Terms Next, we add the two expanded expressions: ( x 3 − 7 x 2 + 4 x ) + ( 2 x 2 − 14 x + 8 ) = x 3 − 7 x 2 + 2 x 2 + 4 x − 14 x + 8 Now, we combine like terms: x 3 + ( − 7 x 2 + 2 x 2 ) + ( 4 x − 14 x ) + 8 = x 3 − 5 x 2 − 10 x + 8
Final Answer The result of the multiplication is x 3 − 5 x 2 − 10 x + 8 . Comparing this with the given options, we see that it matches option B.
Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, in control systems, the transfer function of a system can be represented as a rational function, which involves polynomial multiplication. In computer graphics, polynomial multiplication is used in curve and surface design.
To multiply the polynomials ( x + 2 ) and ( x 2 − 7 x + 4 ) , we distribute terms and combine like terms to get x 3 − 5 x 2 − 10 x + 8 . The correct answer is option B.
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