C. Polynomial and Rational Functions · Polynomial Functions and Equations

The Remainder Theorem

C3.1About 65 minutes

Learning goals

I can explain why the remainder when P(x) is divided by x − b equals P(b).
I can find a remainder without dividing, including for divisors such as 2x + 1.
I can find unknown coefficients from given remainders, including with two unknowns.
I can connect the remainder with the height of the graph of P(x) at x = b.

Words to know

remainder theorem
When a polynomial P(x) is divided by x − b, the remainder is P(b). When it is divided by ax − b, the remainder is P(b/a).

theorem
A statement that has been proved to be true in general.

A shortcut hiding in the division statement

In the last lesson you divided 2x³ + 3x² − 5x + 6 by x + 3 and got a remainder of −6. Now evaluate the polynomial at x = −3:

P(−3) = 2(−27) + 3(9) − 5(−3) + 6 = −54 + 27 + 15 + 6 = −6.

The same number! That is not luck. It is the remainder theorem:

When P(x) is divided by x − b, the remainder is P(b).

The proof takes one line. Start from the division statement and substitute x = b.

Why the remainder theorem works: substituting x = b makes the factor (x − b) equal to 0, wiping out the quotient term and leaving only the remainder.

Dividing by ax − b

The same idea works when the divisor is ax − b. Write P(x) = (ax − b)Q(x) + R and choose the x that makes ax − b = 0, which is x = b/a. Then

the remainder when P(x) is divided by ax − b is P(b/a).

For example, the remainder for 2x + 1 is P(−½), and for 3x − 2 it is P(2/3).

The remainder when P(x) = x³ − 4x² + 2x + 5 is divided by x − 3 is P(3) = 2: the height of the graph above x = 3.

Worked example

Example 1: Remainder without dividing

Find the remainder when x³ − 4x² + 2x + 5 is divided by x − 3. Check with synthetic division.

b = 3: P(3) = 27 − 36 + 6 + 5 = 2.
Check: synthetic division with 3 on 1, −4, 2, 5 gives 1, −1, −1 | 2 ✓.

Remainder 2.

Worked example

Example 2: A divisor like 2x + 1

Find the remainder when P(x) = 2x⁴ + 3x³ − x + 4 is divided by 2x + 1.

2x + 1 = 0 when x = −½, so the remainder is P(−½).
P(−½) = 2(1/16) + 3(−1/8) − (−½) + 4 = 1/8 − 3/8 + 4/8 + 32/8 = 34/8.

Remainder 17/4.

Worked example

Example 3: Finding an unknown coefficient

When x³ + kx² − 2x + 3 is divided by x + 2, the remainder is 7. Find k.

By the remainder theorem, P(−2) = 7.
P(−2) = −8 + 4k + 4 + 3 = 4k − 1.
4k − 1 = 7, so 4k = 8.

k = 2

Worked example

Example 4: Two unknowns

P(x) = x³ + ax² + bx − 6 leaves a remainder of −4 when divided by x − 1 and a remainder of 8 when divided by x + 2. Find a and b.

P(1) = −4: 1 + a + b − 6 = −4, so a + b = 1.
P(−2) = 8: −8 + 4a − 2b − 6 = 8, so 4a − 2b = 22, or 2a − b = 11.
Add the two equations: 3a = 12, so a = 4. Then b = 1 − 4 = −3.
Check: P(x) = x³ + 4x² − 3x − 6. P(1) = 1 + 4 − 3 − 6 = −4 ✓; P(−2) = −8 + 16 + 6 − 6 = 8 ✓.

a = 4, b = −3

Which method when?

To find only the remainder, substitute: it is quickest, especially with a calculator. To find the quotient as well, use synthetic or long division. To check with technology, graph y = P(x) and read the y-value at x = b, or use a CAS command such as GeoGebra's Division.

Just for fun

The remainder theorem is the only place in math class where the leftovers are the main course.

Hands-on activity

Race: divide or substitute?

You need: a partner (a parent, sibling or friend), two calculators, paper and a timer (a phone works).

Problems: find the remainder when x⁴ − 3x³ + 2x − 7 is divided by (a) x − 2, (b) x + 1, (c) x − 5, (d) x + 3, (e) 2x − 1.

One person uses synthetic division for every problem; the other uses the remainder theorem (substituting, with a calculator if wanted).
Time both. Then swap methods with a new polynomial of your own.
Talk about it: when would you still want to divide?

What to look for: the answers are (a) P(2) = 16 − 24 + 4 − 7 = −11, (b) P(−1) = 1 + 3 − 2 − 7 = −5, (c) P(5) = 625 − 375 + 10 − 7 = 253, (d) P(−3) = 81 + 81 − 6 − 7 = 149, (e) P(½) = 1/16 − 3/8 + 1 − 7 = −101/16. Substituting is usually faster; dividing is needed when you also want the quotient.

Your turn

Find the remainder when x³ + 2x² − 5x + 1 is divided by x − 2.

Find the remainder when 2x³ − x² + 3 is divided by x + 1.

Find the remainder when x⁴ − 3x² + x is divided by x + 2.

Find the remainder when 4x³ + 2x − 5 is divided by 2x − 1.

When x³ − kx + 4 is divided by x − 2, the remainder is 2. Find k.

Explain why the remainder when P(x) is divided by x is the constant term of P(x).

Find the remainder when x¹⁰⁰ − 1 is divided by x + 1.

More practice: warm-up

By the remainder theorem, the remainder of P(x) ÷ (x − 4) is ...

The remainder of P(x) ÷ (x + 5) is ...

Find the remainder of (x² + 1) ÷ (x − 1).

The remainder of P(x) ÷ (3x − 2) is ...

Find the remainder of 5x³ ÷ (x − 1).

More practice: on your own

P(x) = 2x³ + kx² + x − 6 leaves a remainder of 4 when divided by x − 2. Find k.

P(x) = x³ + ax² + bx + 2 leaves remainder 3 when divided by x − 1 and remainder −13 when divided by x + 3. Find a and b.

Find the remainder when 6x³ − x² + 2 is divided by 3x + 1.

Find k so that x³ + kx + 1 leaves the same remainder when divided by x − 1 and by x + 1.

The graph of a polynomial P(x) passes through (4, −3). What is the remainder when P(x) is divided by x − 4? Explain.

Challenge

When P(x) is divided by (x − 1)(x − 2), the remainder has the form ax + b. If P(1) = 3 and P(2) = 7, find the remainder.

Show that x − 2 divides x⁵ − 32 evenly, and find the quotient.

Prove that the remainder when P(x) is divided by ax − b (a ≠ 0) is P(b/a).

Check your understanding

Quiz

What is the remainder when x³ − 2x + 5 is divided by x + 2?

Which value gives the remainder when P(x) is divided by 3x + 2?

x³ + kx² − 4 leaves a remainder of 5 when divided by x − 1. What is k?

The graph of y = P(x) passes through (−3, 10). What is the remainder when P(x) is divided by x + 3?

What is the remainder when 2x⁴ − x + 1 is divided by 2x − 1?

P(x) = x³ + ax + b has P(1) = 2 and P(−1) = 6. What are a and b?

For parents and teachers

Big idea (C3.1): dividing P(x) by x − b leaves the remainder P(b). This turns a division problem into a substitution problem, and it is the key to the factor theorem in the next lesson.

How to help: Ask your teen to explain the one-line proof (substitute x = b into P(x) = (x − b)Q(x) + R). If they can explain it, they understand why the theorem works, not just how to use it. Checking a remainder by synthetic division is a good habit.

Common difficulties:

Substituting the wrong sign (for x + 2, use x = −2).
For 2x − 1, substituting x = 1 or x = 2 instead of x = ½.
Arithmetic with fractions and negative powers: (−½)⁴ = 1/16 but (−½)³ = −1/8.
Setting up two-unknown problems but not checking the answer in both conditions.