The polynomial \[ \begin{align} p(x) &= x^4 - 4x^3 + ax^2 + bx - 6 \end{align} \ \] leaves a remainder of 6 when divided by x + 1. If x - 2 is a factor, find the values of a and b.
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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.
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The polynomial \[ \begin{align} p(x) &= x^3 + ax^2 + bx - 4 \quad \end{align} \ \] has \( (x^2 - 4) \) as a factor. Find the values of a and b and then factor p(x).
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Find a polynomial p(x), with integer coefficients whose zeroes are \[ \begin{align} \frac{1}{2}, -\frac{1}{3}, -7 \text{ and} \quad 5 \end{align} \ \]
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