Question: Question 3 - of 7 Step 1 of 1 Use synthetic division to determine if the given value for \( k \)…

Question 3 - 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 \( p(k) \).

\[ p(x) = 12x^4 - 41x^3 - 9x^2 + 82x - 30; k = 3 \]

Answer 6 Points

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

• Yes
• No

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Solution

Step 1: List the coefficients of the polynomial \( p(x) \). \[ 12 \quad -41 \quad -9 \quad 82 \quad -30 \] Step 2: Set up the synthetic division with \( k = 3 \). \[ \begin{array}{c|ccccc} 3 & 12 & -41 & -9 & 82 & -30 \\ & & 36 & -15 & -72 & 30 \\ \hline & 12 & -5 & -24 & 10 & 0 \\ \end{array} \] Step 3: Analyze the remainder. The remainder is \( 0 \). Step 4: Conclude whether \( k = 3 \) is a zero of the polynomial. Yes, \( k = 3 \) is a zero of the polynomial \( p(x) \).