3.4 The Remainder and Factor Theorems

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Notes

Remainder Theorem: when $p(x)$ is divided by $x - c$, the remainder is $p(c)$.
Factor Theorem: $x - c$ is a factor of $p(x)$ exactly when $p(c) = 0$.
So to test a factor $x + 3$, check whether $p(-3) = 0$.
Example: $p(x) = x^2 - 3x - 4$ gives $p(4) = 0$, so $x - 4$ is a factor.

Find the remainder when $x^3 - 4x^2 + 7$ is divided by $x - 2$.

Is $x + 3$ a factor of $p(x) = x^3 + 3x^2 - 4x - 12$?

For what value of $k$ is $x - 1$ a factor of $x^3 + kx^2 - 3x + 5$?