Questions: Use the remainder theorem to determine if the given number c is a zero of the polynomial. f(x)=4x^4+2x^3-17x^2+17x+54 (a) c=7 (Choose one) a zero of the polynomial. (b) c=-2 (Choose one) a zero of the polynomial.

Use the remainder theorem to determine if the given number c is a zero of the polynomial. f(x)=4x^4+2x^3-17x^2+17x+54 (a) c=7 (Choose one) a zero of the polynomial. (b) c=-2 (Choose one) a zero of the polynomial.
Transcript text: Use the remainder theorem to determine if the given number $c$ is a zero of the polynomial. \[ f(x)=4 x^{4}+2 x^{3}-17 x^{2}+17 x+54 \] (a) $c=7$ (b) $c=-2$ Part 1 of 2 (a) $c=7$ $\square$ (Choose one) a zero of the polynomial. Part 2 of 2 (b) $c=-2$ (Choose one) $\square$ a zero of the polynomial.
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Solution

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Solution Steps

To determine if a given number \( c \) is a zero of the polynomial \( f(x) \), we can use the Remainder Theorem. According to the theorem, if \( f(c) = 0 \), then \( c \) is a zero of the polynomial. We will evaluate the polynomial at the given values of \( c \) and check if the result is zero.

Step 1: Evaluate \( f(7) \)

To determine if \( c = 7 \) is a zero of the polynomial \( f(x) = 4x^4 + 2x^3 - 17x^2 + 17x + 54 \), we evaluate \( f(7) \):

\[ f(7) = 4(7^4) + 2(7^3) - 17(7^2) + 17(7) + 54 \]

Calculating each term:

  • \( 7^4 = 2401 \) so \( 4(7^4) = 9604 \)
  • \( 7^3 = 343 \) so \( 2(7^3) = 686 \)
  • \( 7^2 = 49 \) so \( -17(7^2) = -833 \)
  • \( 17(7) = 119 \)
  • The constant term is \( 54 \)

Adding these results together: \[ f(7) = 9604 + 686 - 833 + 119 + 54 = 10000 + 54 - 833 = 10000 - 779 = 9221 \]

Since \( f(7) \neq 0 \), we conclude that \( c = 7 \) is not a zero of the polynomial.

Step 2: Evaluate \( f(-2) \)

Next, we check if \( c = -2 \) is a zero of the polynomial by evaluating \( f(-2) \):

\[ f(-2) = 4(-2^4) + 2(-2^3) - 17(-2^2) + 17(-2) + 54 \]

Calculating each term:

  • \( -2^4 = 16 \) so \( 4(-2^4) = 64 \)
  • \( -2^3 = -8 \) so \( 2(-2^3) = 16 \)
  • \( -2^2 = 4 \) so \( -17(-2^2) = 68 \)
  • \( 17(-2) = -34 \)
  • The constant term is \( 54 \)

Adding these results together: \[ f(-2) = 64 + 16 + 68 - 34 + 54 = 64 + 16 + 68 + 20 = 168 \]

Since \( f(-2) = 0 \), we conclude that \( c = -2 \) is a zero of the polynomial.

Final Answer

  • For part (a), \( c = 7 \) is not a zero of the polynomial.
  • For part (b), \( c = -2 \) is a zero of the polynomial.

Thus, the answers are:

  • (a) \( \boxed{\text{not a zero}} \)
  • (b) \( \boxed{\text{a zero}} \)
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