Remainder and Factor Theorems

The polynomial \[ \begin{align} p(x) &= x^4 + x^3 - 11x^2 - 9x + 2k^2 \end{align} \ \] (a) Find the value of k for which p(x) is divisible by both (x - 1) and (x - 3)
(b) Solve p(x) = 0 for this value of k.

The value of k must be a common solution to p(1) = 0 and p(3) = 0.

Solving p(1) = 0: \[ \begin{align} 1 + 1 - 11 - 9 + 2k^2 &= 0 \\ 2k^2 - 9 - 9 &= 0 \\ 2k^2 &= 18 \end{align} \ \] So k = -3 or +3.

Solving p(3) = 0: \[ \begin{align} 81 + 27 - 99 - 27 + 2k^2 &= 0 \\ 2k^2 - 18 &= 0 \\ k^2 &= 9 \end{align} \ \] So k = -3 or +3 again.

Test the factors of the constant term to see if they satisfy p(x).

Setting k = 3, \[ \begin{align} p(-2) &= 16 - 8 - 44 + 18 + 18 \\ &= 0 \\ p(-3) &= 81 - 27 - 99 + 27 + 18 \\ &= 0 \end{align} \ \] So (x+2) and (x+3) are factors, and \[ \begin{align} p(x) &= (x -1)(x + 2)(x - 3)(x + 3) \quad \quad \quad \quad \end{align} \ \]

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What can you say about k?

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Now You Try:

The polynomial \[ \begin{align} p(x) &= 4x^3 - x + k \end{align} \ \] (a) Find the value of k for which p(x) is divisible by (2x+3)
(b) For this value of k, find the other factor.