Question

select the correct answer.
one factor of the polynomial ( 3x^3 + 20x^2 - 21x + 88 ) is ( (x + 8) ). what is the other factor of the polynomial? (note: use long division )

a. ( (3x^2 - 4x + 11) )

b. ( (3x^2 - 4x) )

c. ( (3x^2 + 11) )

d. ( (3x^2 + 4x - 11) )

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(3x^3\) by the leading term of the divisor \(x\), we get \(3x^2\). Multiply the divisor \((x + 8)\) by \(3x^2\) to get \(3x^3+24x^2\). Subtract this from the dividend: \((3x^3 + 20x^2-21x + 88)-(3x^3+24x^2)=-4x^2-21x + 88\).

Step2: Divide the new leading terms

Divide the leading term of the new dividend \(-4x^2\) by the leading term of the divisor \(x\), we get \(-4x\). Multiply the divisor \((x + 8)\) by \(-4x\) to get \(-4x^2-32x\). Subtract this from the new dividend: \((-4x^2-21x + 88)-(-4x^2-32x)=11x + 88\).

Step3: Divide the new leading terms

Divide the leading term of the new dividend \(11x\) by the leading term of the divisor \(x\), we get \(11\). Multiply the divisor \((x + 8)\) by \(11\) to get \(11x+88\). Subtract this from the new dividend: \((11x + 88)-(11x + 88)=0\).
So the quotient is \(3x^2-4x + 11\), which is the other factor.

Answer:

A. \((3x^{2}-4x + 11)\)