Watch the video and then solve the problem given below. Click here to watch the video. Using synthetic division, determine whether the numbers are zeros of the polynomial function. \( -3,2 ; f(x)=2 x^{3}+3 x^{2}-5 x+12 \) Choose the correct answer below. A. Both the numbers -3 and 2 are zeros of the polynomial function as \( f(-3)=0 \) and \( f(2)=0 \). B. The number -3 is not a zero of the polynomial function as \( f(-3)=\square \) and the number 2 is a zero of the polynomial function as \( =\square \). C. The number -3 is a zero of the polynomial function as \( f(-3)=\square \) and the number 2 is not a zero of the polynomial function as \( f(2)=\square \). D. Neither -3 nor 2 is a zero of the polynomial function as \( f(-3) \neq 0 \) and \( f(2) \neq 0 \).
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The Deep Dive
When using synthetic division to check if the numbers -3 and 2 are zeros of the polynomial \( f(x) = 2x^3 + 3x^2 - 5x + 12 \), you will find that substituting -3 and 2 into the polynomial yields non-zero results. This means that neither -3 nor 2 are roots of the polynomial, confirming option D as the correct answer. Synthetic division is a streamlined method that saves time when evaluating polynomials. To perform synthetic division, you set up your coefficients and execute the division process, dropping down each coefficient while multiplying and adding along the way. It's a fun little trick that makes polynomial evaluation super efficient, and a great mental math exercise!
