Question

select the correct answer
using the remainder theorem, what is the remainder of the quotient of ( x^3 - 6x^2 + 4x - 5 ) and ( x - 3 )?
a. 34
b. -74
c. -20
d. 88

Explanation:

Step1: Recall Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). Here, \( f(x)=x^{3}-6x^{2}+4x - 5 \) and \( a = 3 \) (since the divisor is \( x - 3 \)).

Step2: Substitute \( x = 3 \) into \( f(x) \)

Calculate \( f(3)=(3)^{3}-6(3)^{2}+4(3)-5 \).
First, \( 3^{3}=27 \), \( 6(3)^{2}=6\times9 = 54 \), \( 4(3)=12 \).
Then, \( f(3)=27-54 + 12-5 \).
Simplify: \( 27-54=-27 \), \( -27 + 12=-15 \), \( -15-5=-20 \).

Answer:

C. -20