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

Solution

Step 1 Set up the synthetic division with \( k = 3 \). \[ \begin{array}{ccccc} 3 & -13 & -8 & 75 & -45 \\ \end{array} \] Step 2 Bring down the first coefficient. \[ 3 \] Step 3 Multiply by \( k \) and add to the next coefficient. \[ 3 \times 3 = 9 \] \[ -13 + 9 = -4 \] Step 4 Repeat the process. \[ -4 \times 3 = -12 \] \[ -8 + (-12) = -20 \] Step 5 Continue the synthetic division. \[ -20 \times 3 = -60 \] \[ 75 + (-60) = 15 \] Step 6 Finalize the synthetic division. \[ 15 \times 3 = 45 \] \[ -45 + 45 = 0 \] Step 7 Since the remainder is \( 0 \), \( k = 3 \) is a zero of the polynomial. Answer: Yes.