The remainder and factor theorems

The remainder theorem and the factor theorem with proofs, finding integer zeroes from factors of the constant term, and factoring a polynomial fully.

The remainder theorem

Example 1●

Find the remainder when $P(x) = x^3 - 4x^2 + 2x + 5$ is divided by $x + 1$.

$P(-1) = -1 - 4 - 2 + 5 = -2$

x + 1 = x − (−1), so a = −1.

This is the remainder theorem with remainder zero. It turns "is this a factor?" into a one-line calculation.

Remainders from the graph

The remainder on dividing by x − a is the height of the graph at x = a. Drag A onto each x-intercept.

Try this

Drag $A$ to each of the three intercepts. What are the three linear factors?

Example 2●●

Factor $P(x) = x^3 - 2x^2 - 5x + 6$ fully.

$P(1) = 1 - 2 - 5 + 6 = 0$, so $(x - 1)$ is a factor.

Candidates are the factors of 6: ±1, ±2, ±3, ±6.

$P(x) = (x - 1)(x^2 - x - 6)$

Divide by x − 1, or match coefficients of (x − 1)(x² + bx + c).

$P(x) = (x - 1)(x - 3)(x + 2)$

Example 3: an unknown coefficient●●●

$x - 2$ is a factor of $P(x) = x^3 + kx^2 - 4x + 12$. Find $k$, then the other zeroes.

$P(2) = 8 + 4k - 8 + 12 = 0$, so $k = -3$.

$x^3 - 3x^2 - 4x + 12 = x^2(x - 3) - 4(x - 3) = (x - 3)(x - 2)(x + 2)$

The other zeroes are $3$ and $-2$.

Flashcards

Remainder theorem

Dividing $P(x)$ by $x - a$ leaves remainder $P(a)$.

Factor theorem

$(x - a)$ is a factor exactly when $P(a) = 0$.

Integer zeroes of a monic polynomial

Must divide the constant term.

Practice