Question

use synthetic division to determine if the given value for k is a zero of this polynomial. if not, determine p(k).
p(x) = 4x⁴ - 12x³ - 8x² + 46x - 28; k = 2

answer
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is k a zero of this polynomial?
○ yes ○ no

Explanation:

Step1: Set up synthetic division

The polynomial is \( p(x) = 4x^4 - 12x^3 - 8x^2 + 46x - 28 \) and \( k = 2 \). For synthetic division, we use the coefficients of the polynomial: \( 4, -12, -8, 46, -28 \) (note that the polynomial is in standard form, so we have all degrees from 4 down to 0, with the coefficient of \( x^0 \) being -28).
Set up the synthetic division as follows:
Bring down the first coefficient:

$$
LATEXBLOCK0
$$

Step2: Multiply and add

Multiply the brought - down coefficient (4) by \( k = 2 \): \( 4\times2 = 8 \). Add this to the next coefficient: \( -12+8=-4 \)

$$
LATEXBLOCK1
$$

Multiply - 4 by 2: \( -4\times2=-8 \). Add to the next coefficient: \( -8+( - 8)=-16 \)

$$
LATEXBLOCK2
$$

Multiply - 16 by 2: \( -16\times2 = - 32 \). Add to the next coefficient: \( 46+( - 32)=14 \)

$$
LATEXBLOCK3
$$

Multiply 14 by 2: \( 14\times2 = 28 \). Add to the last coefficient: \( -28 + 28=0 \)

$$
LATEXBLOCK4
$$

The remainder is 0. By the Remainder Theorem, if the remainder when a polynomial \( p(x) \) is divided by \( (x - k) \) is 0, then \( k \) is a zero of the polynomial.

Answer:

Yes