Remainder and Factor Theorems

Find a polynomial p(x), with integer coefficients whose zeroes are \[ \begin{align} \frac{1}{2}, -\frac{1}{3}, -7 \text{ and} \quad 5 \end{align} \ \]

Find a general expression for p(x)

A general expression for p(x) is \[ \begin{align} p(x) &= k(x-1/2)(x+1/3)(x+7)(x-5) \end{align} \ \]

To guarantee integer coefficients, rewrite the expression for p(x)

\[ \begin{align} p(x) &= k(\frac{2x-1}{2})(\frac{3x+1}{3})(x+7)(x-5) \quad \\ &= \frac{k}{6}(2x-1)(3x+1)(x+7)(x-5) \end{align} \ \] So k must be a multiple of 6.

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

y

Now You Try:

Find a polynomial p(x), with integer coefficients whose zeroes are \[ \begin{align} \frac{1}{2}, -\frac{1}{3} \text{ and} \quad 2 \end{align} \ \]