The factor and remainder theorem - Top Study Guide

Remainder theorem when we divide a polynomial f(x) by x − c the remainder r equals f(c)

Let’s say

f (x) ÷ (x − c) = q(x) + r

where r is the remainder. We also know

f (x) = (x − c)q(x) + r

If we now substitute x with c

f (c) = (c − c)q(c) + r

but c − c = 0, therefore

f (c) = r

Factor theorem when f (c) = 0 then x − c is a factor of the polynomial

Frequently Asked Questions: Remainder and Factor Theorems

What is the Remainder Theorem?

The Remainder Theorem states that if a polynomial P(x) is divided by a linear factor (x - c), then the remainder of that division is equal to P(c).

In simpler terms: To find the remainder when a polynomial is divided by (x - c), you don't need to do long division. Just substitute c into the polynomial.

What is the Factor Theorem?

The Factor Theorem is a direct consequence of the Remainder Theorem. It states that a polynomial P(x) has a factor (x - c) if and only if P(c) = 0.

This means if substituting c into the polynomial gives you zero, then (x - c) divides the polynomial exactly with no remainder, making it a factor.

How are the Remainder Theorem and Factor Theorem related or similar?

They are very closely related! The Factor Theorem is essentially a special case of the Remainder Theorem. The Remainder Theorem tells you the value of the remainder (which is P(c)). The Factor Theorem uses this to identify when the remainder is zero, which is the condition for (x - c) to be a factor.

They both rely on evaluating the polynomial at a specific value related to the divisor (x - c).

What is the difference between the Remainder Theorem and Factor Theorem?

The difference lies in what they tell you:

Remainder Theorem: Tells you the exact value of the remainder when dividing by (x - c) (the remainder is P(c)).
Factor Theorem: Tells you *if* the divisor (x - c) is a factor (it is a factor if and only if the remainder, P(c), is 0).

One tells you *what* the remainder is, the other tells you *whether* the remainder is zero (and thus *whether* it's a factor).

How do you use the Remainder Theorem to find a remainder?

To find the remainder when dividing a polynomial P(x) by (x - c):

Identify the value c from the divisor (x - c). (If the divisor is (x + c), then the value is -c).
Substitute this value c into the polynomial P(x) to find P(c).
The result, P(c), is the remainder.

Example: Find the remainder of P(x) = x² - 5x + 6 when divided by (x - 2). Here, c = 2. Evaluate P(2) = (2)² - 5(2) + 6 = 4 - 10 + 6 = 0. The remainder is 0.

How do you use the Factor Theorem to find factors?

To use the Factor Theorem to check if (x - c) is a factor of P(x), or to find potential factors:

If checking a specific (x - c), identify c and calculate P(c). If P(c) = 0, then (x - c) is a factor.
To find potential rational factors (x - c) of a polynomial with integer coefficients, look at possible values for c. According to the Rational Root Theorem (which often pairs with the Factor Theorem), if c = p/q is a rational root, then p must be a divisor of the constant term of P(x) and q must be a divisor of the leading coefficient. Test these potential c values by evaluating P(c). If P(c) = 0, then (x - c) is a factor.

Example: Is (x - 3) a factor of P(x) = x² - 5x + 6? Here, c = 3. Evaluate P(3) = (3)² - 5(3) + 6 = 9 - 15 + 6 = 0. Since P(3) = 0, (x - 3) is a factor.

How do you factorize a polynomial using the Remainder/Factor Theorem?

The Factor Theorem helps you *find* factors. Once you find a factor (x - c) because you've shown P(c) = 0, you can then use polynomial long division or synthetic division to divide P(x) by (x - c). The result will be a polynomial of lower degree. You can then continue trying to factor the resulting polynomial using the same theorems or other factoring techniques until the polynomial is fully factored.

So, the theorems help identify roots/factors, and division helps break down the polynomial using those factors.