3.1 Remainder and Factor Theorems

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What is a Polynomial?

A polynomial is an expression made up of terms with non-negative whole-number powers of $x$. For example:

$P(x) = 2x^3 - 5x^2 + 3x - 7$

This is called a cubic polynomial because the highest power is 3. We write $P(x)$ (read as "P of x") to show it's a function — meaning we can substitute any value of $x$ into it.

Part 1 — The Remainder Theorem

The Big Idea

Normally, when you divide one number by another, you might get a remainder. For example, $17 \div 5 = 3$ remainder $2$.

The same thing happens with polynomials. When you divide a polynomial $P(x)$ by a linear factor (an expression like $x - c$), you usually get a quotient (the main result) and a remainder (what's left over).

The Remainder Theorem gives you a shortcut to find that remainder — without doing any long division at all.

The Remainder Theorem — Statement

In other words: to find the remainder, just substitute $x = c$ into the polynomial.

Where does this come from?

When $P(x)$ is divided by $(x - c)$, we can write:

$P(x) = (x - c) \cdot Q(x) + R$

Here, $Q(x)$ is the quotient and $R$ is the remainder (a number, not an expression).

Now substitute $x = c$:

$P(c) = (c - c) \cdot Q(c) + R = 0 \cdot Q(c) + R = R$

So $P(c) = R$. The remainder is simply the value of the polynomial at $x = c$.

Worked Example 1 — Basic Remainder

Find the remainder when $7x^3 + 6x^2 - 40x + 17$ is divided by $(x + 3)$.

Step 1: Write the divisor in the form $(x - c)$.

$(x + 3) = (x - (-3))$, so $c = -3$.

Step 2: Substitute $x = -3$ into the polynomial.

$P(-3) = 7(-3)^3 + 6(-3)^2 - 40(-3) + 17$

$= 7(-27) + 6(9) + 120 + 17$

$= -189 + 54 + 120 + 17$

$= \mathbf{2}$

✅ The remainder is 2.