Remainder and Factor Theorems

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.

Find two equations in the unknowns a and b.

From the remainder theorem, \[ \begin{align} p(-1) &= (-1)^4 - 4(-1)^3 + a(-1)^2 + b(-1) - 6 \quad \\ 6 &= 1 + 4 + a - b - 6 \\ a - b &= 7 \quad \quad (1) \end{align} \ \] From the factor theorem, \[ \begin{align} p(2) &= (2)^4 - 4(2)^3 + a(2)^2 + b(2) - 6 \quad \\ 0 &= 16 - 32 + 4a + 2b - 6 \\ 2a + b &= 19 \quad \quad (2) \end{align} \ \]

Eliminate b to find a first.

Adding (1) and (2): \[ \begin{align} 3a &= 26 \\ a &= 26/3 \end{align} \ \] from (1), \[ \begin{align} 26/3 - b &= 7 \\ b &= 26/3 - 7 \\ &= 5/3 \end{align} \ \] So \[ p(x) = x^4 - 4x^3 + \frac{26}{3}x^2 + \frac{5}{3}x - 6 \]

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

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Now You Try:

The polynomial \[ \begin{align} p(x) &= 3x^3 + ax^2 + x + b \end{align} \ \] leaves a remainder of -60 when divided by x + 2. If x - 1 is a factor, find the values of a and b.