Question

given $p(x)=x^3 + 4x^2 - 3x + 2$ and $\frac{p(x)}{x - 1}=x^2 + 5x + 2$ r4. which statement is true? select all that apply. a $p(x)=(x - 1)cdot(x^2 + 5x + 2)+4$ b $p(x)=(x - 1)cdot(x^2 + 5x + 2)+\frac{4}{x - 1}$ c $p(1)=4$ d $p(-1)=4$

Explanation:

To determine the true statements, we use the division algorithm for polynomials: when dividing a polynomial \( p(x) \) by \( (x - a) \), we have \( p(x)=(x - a)q(x)+r \), where \( q(x) \) is the quotient and \( r \) is the remainder (a constant since the divisor is linear).

• For Option A: Given \( \frac{p(x)}{x - 1}=x^{2}+5x + 2 \) with remainder \( 4 \), by the division algorithm, \( p(x)=(x - 1)(x^{2}+5x + 2)+4 \). This matches the formula, so A is true.
• For Option B: The remainder in polynomial division is a constant (not a fraction with \( x - 1 \) in the denominator) because the divisor \( (x - 1) \) is linear. So B is false.
• For Option C: Substitute \( x = 1 \) into \( p(x)=(x - 1)(x^{2}+5x + 2)+4 \). The first term \( (1 - 1)(\dots)=0 \), so \( p(1)=0 + 4=4 \). Thus, C is true.
• For Option D: Substitute \( x=-1 \) into \( p(x)=x^{3}+4x^{2}-3x + 2 \): \( p(-1)=(-1)^{3}+4(-1)^{2}-3(-1)+2=-1 + 4 + 3 + 2=8

eq4 \). So D is false.

Answer:

A. \( p(x)=(x - 1)\cdot(x^{2}+5x + 2)+4 \)
C. \( p(1)=4 \)