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Identify the polynomial and apply the Factor Theorem.

Mathematics

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Answer

-4\text{-4}

Step-by-step solution

4 steps

  1. Identify the polynomial and apply the Factor Theorem.

    The polynomial is P(x)=2x3−8x+x2+kP(x) = 2x^3 - 8x + x^2 + k. It's good practice to write it in descending powers of xx: P(x)=2x3+x2−8x+kP(x) = 2x^3 + x^2 - 8x + k.
    The given factor is (2x+1)(2x+1).
    According to the Factor Theorem, if (2x+1)(2x+1) is a factor of P(x)P(x), then P(−12)P\left(-\frac{1}{2}\right) must be equal to 0.

  2. Substitute x=−12x = -\frac{1}{2} into the polynomial and set it equal to 0.

    P(−12)=2(−12)3+(−12)2−8(−12)+k=0P\left(-\frac{1}{2}\right) = 2\left(-\frac{1}{2}\right)^3 + \left(-\frac{1}{2}\right)^2 - 8\left(-\frac{1}{2}\right) + k = 0

  3. Simplify the terms.

    2(−18)+(14)−8(−12)+k=02\left(-\frac{1}{8}\right) + \left(\frac{1}{4}\right) - 8\left(-\frac{1}{2}\right) + k = 0 −28+14+82+k=0-\frac{2}{8} + \frac{1}{4} + \frac{8}{2} + k = 0 −14+14+4+k=0-\frac{1}{4} + \frac{1}{4} + 4 + k = 0

  4. Solve for kk.

    0+4+k=00 + 4 + k = 0 4+k=04 + k = 0 k=−4k = -4

    The value of the constant kk is -4\boxed{\text{-4}}.

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