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If (x-5) is a factor of x3 - 4x2 - 11x + 30, find the remaining factors.

Mathematics

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Answer

C

Step-by-step solution

3 steps

3. If (x−5)(x-5) is a factor of x3−4x2−11x+30x^3 - 4x^2 - 11x + 30, find the remaining factors.

  1. Since (x−5)(x-5) is a factor, we can use synthetic division with x=5x=5 to divide the polynomial x3−4x2−11x+30x^3 - 4x^2 - 11x + 30.

    51−4−113055−3011−60\begin{array}{c|cccc} 5 & 1 & -4 & -11 & 30 \\ & & 5 & 5 & -30 \\ \hline & 1 & 1 & -6 & 0 \\ \end{array}

    The coefficients of the quotient are 1,1,−61, 1, -6, and the remainder is 0.
    This means the quotient is x2+x−6x^2 + x - 6.

  2. Factor the quadratic expression obtained from the division.

    We need to factor x2+x−6x^2 + x - 6. We look for two numbers that multiply to −6-6 and add to 1. These numbers are 3 and −2-2.
    So, the quadratic factors as:

    x2+x−6=(x+3)(x−2)x^2 + x - 6 = (x+3)(x-2)
  3. Identify the remaining factors.

    The remaining factors are (x+3)(x+3) and (x−2)(x-2).

    Comparing this with the given options:
    A. (x−3)(x-3) and (x+2)(x+2)
    B. (x−3)(x-3) and (x−2)(x-2)
    C. (x+3)(x+3) and (x−2)(x-2)
    D. (x+3)(x+3) and (x+2)(x+2)

    The correct option is C.

    The final answer is C\boxed{\text{C}}

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