Question

1. what is remainder of the quotient:

$(2x^3 - 5x^2 + 3x - 4) \div (x - 2)$
2 points (a.apr.d.6)

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, we are dividing by \( (x - 2) \), so \( a = 2 \), and we need to find \( f(2) \) where \( f(x)=2x^{3}-5x^{2}+3x - 4 \).

Step2: Substitute \( x = 2 \) into \( f(x) \)

Calculate each term:

• For \( 2x^{3} \): \( 2\times(2)^{3}=2\times8 = 16 \)
• For \( -5x^{2} \): \( -5\times(2)^{2}=-5\times4=-20 \)
• For \( 3x \): \( 3\times2 = 6 \)
• The constant term: \( -4 \)

Now, sum these terms: \( f(2)=16-20 + 6-4 \)

Step3: Simplify the expression

First, \( 16-20=-4 \), then \( -4 + 6 = 2 \), then \( 2-4=-2 \).

Answer:

The remainder is \(-2\)