Question: Question 6 - of 7 Step 1 of 1 Use synthetic division to determine if the given value for \( k \)…

Question 6 - of 7 Step 1 of 1

Use synthetic division to determine if the given value for \( k \) is a zero of this polynomial. If not, determine \( p(k) \).

\[ p(x) = 3x^4 - 13x^3 - 8x^2 + 75x - 45; k = 3 \]

Answer 6 Points

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Is \( k \) a zero of this polynomial?

• Yes
• No

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Solution

Set up the synthetic division with \( k = 3 \). \[ 3 \quad -13 \quad -8 \quad 75 \quad -45 \] Bring down the first coefficient: \[ 3 \] Multiply by 3: \[ 3 \times 3 = 9 \] Add to the next coefficient: \[ -13 + 9 = -4 \] Multiply by 3: \[ -4 \times 3 = -12 \] Add to the next coefficient: \[ -8 + (-12) = -20 \] Multiply by 3: \[ -20 \times 3 = -60 \] Add to the next coefficient: \[ 75 + (-60) = 15 \] Multiply by 3: \[ 15 \times 3 = 45 \] Add to the last coefficient: \[ -45 + 45 = 0 \] The remainder is 0, so \( p(3) = 0 \). Answer to the multiple-choice question: Yes.