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A polynomial function has zeros at (3)/(4) and -1 . Which of the following must be a factor of the polynomial? Choose 1 answer: (A) x-1 (B) x+1 (c) 3x-4 (D) 3x+4

A polynomial function has zeros at 34 \frac{3}{4} and −1-1 . Which of the following must be a factor of the polynomial?\newlineChoose 11 answer:\newline(A) x−1 x-1 \newline(B) x+1 x+1 \newline(C) 3x−4 3 x-4 \newline(D) 3x+4 3 x+4

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Full solution

Q. A polynomial function has zeros at 34 \frac{3}{4} and −1-1 . Which of the following must be a factor of the polynomial?\newlineChoose 11 answer:\newline(A) x−1 x-1 \newline(B) x+1 x+1 \newline(C) 3x−4 3 x-4 \newline(D) 3x+4 3 x+4
  1. Zeros of a Polynomial: The zeros of a polynomial are the values of xx for which the polynomial equals zero. If a polynomial has a zero at x=ax = a, then (x−a)(x - a) is a factor of the polynomial.
  2. Finding the Corresponding Factor: Given the zero at x=34x = \frac{3}{4}, we can find the corresponding factor by subtracting this zero from xx: x−34x - \frac{3}{4}.
  3. Expressing the Factor with Integer Coefficients: To express the factor with integer coefficients, we multiply by the denominator of the zero to clear the fraction: 4(x−34)=4x−34(x - \frac{3}{4}) = 4x - 3.
  4. Finding an Equivalent Factor: The factor corresponding to the zero at x=34x = \frac{3}{4} is therefore 4x−34x - 3. However, this factor is not listed in the answer choices. We need to find a factor that is equivalent to one of the given choices.
  5. Dividing by the Greatest Common Divisor: Since we are looking for integer coefficients, we can divide the factor 4x−34x - 3 by the greatest common divisor of its coefficients, which is 11 in this case. This does not change the factor, so we still have 4x−34x - 3.
  6. Considering Another Zero: We notice that the factor 4x−34x - 3 is not a multiple of any of the answer choices. However, if we consider the zero at x=−1x = -1, the corresponding factor is x−(−1)x - (-1) or x+1x + 1.
  7. Matching the Factor to an Answer Choice: The factor x+1x + 1 corresponds to choice (B) in the answer choices. Since the polynomial has a zero at x=−1x = -1, x+1x + 1 must be a factor of the polynomial.

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