Select the correct answer.
One factor of the polynomial $3 x^3+20 x^2-21 x+88$ is $(x+8)$. What is the other factor of the polynomial? (Note: Use long division.)
A. $\left(3 x^2-4 x+11\right)$
B. $\left(3 x^2-4 x\right)$
C. $\left(3 x^2+11\right)$
D. $\left(3 x^2+4 x-11\right)$
Answer (1)
We perform polynomial long division to find the other factor of the polynomial. The polynomial is 3 x 3 + 20 x 2 − 21 x + 88 and one factor is ( x + 8 ) .
Divide 3 x 3 + 20 x 2 − 21 x + 88 by ( x + 8 ) .
The quotient is 3 x 2 − 4 x + 11 .
The other factor of the polynomial is ( 3 x 2 − 4 x + 11 ) .
The correct answer is 3 x 2 − 4 x + 11 .
Explanation
Problem Analysis We are given a polynomial 3 x 3 + 20 x 2 − 21 x + 88 and one of its factors ( x + 8 ) . Our goal is to find the other factor of the polynomial. We can use polynomial long division to divide the given polynomial by the known factor to find the other factor.
Long Division Setup We will perform polynomial long division to divide 3 x 3 + 20 x 2 − 21 x + 88 by ( x + 8 ) .
First Step of Division Dividing 3 x 3 by x gives 3 x 2 . Multiply ( x + 8 ) by 3 x 2 to get 3 x 3 + 24 x 2 . Subtract this from the original polynomial: ( 3 x 3 + 20 x 2 − 21 x + 88 ) − ( 3 x 3 + 24 x 2 ) = − 4 x 2 − 21 x + 88
Second Step of Division Now, divide − 4 x 2 by x to get − 4 x . Multiply ( x + 8 ) by − 4 x to get − 4 x 2 − 32 x . Subtract this from the remaining polynomial: ( − 4 x 2 − 21 x + 88 ) − ( − 4 x 2 − 32 x ) = 11 x + 88
Third Step of Division Next, divide 11 x by x to get 11 . Multiply ( x + 8 ) by 11 to get 11 x + 88 . Subtract this from the remaining polynomial: ( 11 x + 88 ) − ( 11 x + 88 ) = 0
Finding the Other Factor The quotient is 3 x 2 − 4 x + 11 , and the remainder is 0 . Therefore, the other factor of the polynomial is ( 3 x 2 − 4 x + 11 ) .
Final Answer Comparing our result with the given options, we see that the correct answer is A. ( 3 x 2 − 4 x + 11 ) .
Examples
Polynomial factorization is used in various fields such as cryptography, data compression, and error correction. For example, in cryptography, factoring large polynomials is essential for creating secure encryption keys. In data compression, polynomial factorization helps in identifying patterns and redundancies in data, leading to more efficient compression algorithms. In error correction, polynomials are used to encode data in such a way that errors can be detected and corrected during transmission.
