Polynomials path · A03

Before this lesson: Polynomial division, How to write a proof

Your goal: Choose a substitution or a new polynomial that turns given values into factors.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Remainder Theorem and Factor Theorem: the key idea

If P(x)=(x−a)Q(x)+r, substitution x=a makes the product vanish, so r=P(a). This is the remainder theorem. In particular x−a is a factor exactly when P(a)=0. To use several supplied values, choose an auxiliary polynomial: if P(a)=a, then P(x)−x has a zero at a. For a product of distinct linear divisors, write a remainder of low enough degree and use their roots to determine it. Over the integers, the factorisation ak−bka^{k}-b^{k} also shows that a−b divides P(a)−P(b) when P has integer coefficients and a≠b.

A worked example

P(−1)=4 and P(2)=1. Find the remainder on division by (x+1)(x−2).

The divisor is quadratic, so write the remainder as ux+v. Evaluating at its two distinct roots gives −u+v=4 and 2u+v=1. Subtracting yields 3u=−3, so u=−1 and v=3. The remainder is 3−x; the entire polynomial P need not be linear.

Your turn: change one thing

Find k if x−2 divides x³+kx+6.

Try this on paper before opening the explanation.

Compare your reasoning

Substitute 2 and set the value to zero: 8+2k+6=0. Thus k=−7. Check x³−7x+6=(x−2)(x−1)(x+3).

Pause and check

A trap to avoid: Confusing P(a)=a with P(a)=0; the integer-coefficient extension also requires N01 and N07.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove that for an integer-coefficient polynomial P and distinct integers a,b, a−b divides P(a)−P(b).

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

For k≥1, ak−bk=(a−b)(ak−1+ak−2b+…+bk−1)a^{k}-b^{k}=(a-b)(a^{k-1}+a^{k-2}b+\ldots +b^{k-1}). The second factor is an integer. In P(a)−P(b), the constant terms cancel and every other term is an integer coefficient times one of these differences. Each is divisible by a−b, so their sum is too.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

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

Foundation

  1. 3
  2. 5
  3. 7
  4. 6
Hint

Substitute the root 1 of the divisor.

Answer and reasoning

6. P(1)=1+2+3=6.

2. Which condition is equivalent to x+3 dividing P(x)?

Foundation

  1. P(−3)=3
  2. P(−3)=0
  3. P(3)=0
  4. P(0)=3
Hint

The root of x+3 is −3.

Answer and reasoning

P(−3)=0. The factor theorem uses the input that makes the linear factor zero.

3. If x−1 divides x²+kx−5, find k.

Core

  1. 5
  2. 4
  3. -4
  4. 3
Hint

Set P(1)=0.

Answer and reasoning

4. 1+k−5=0, so k=4.

4. P(0)=2 and P(1)=5. What is its remainder modulo x(x−1)?

Core

  1. 3
  2. 3x+2
  3. 2x+3
  4. 5x+2
Hint

Write a linear remainder and use both values.

Answer and reasoning

3x+2. A remainder ux+v has v=2 and u+v=5, giving u=3.

5. If P(1)=1 and P(2)=2, which polynomial is divisible by (x−1)(x−2)?

Stretch

  1. P(x)
  2. P(x)+x
  3. P(x)−1
  4. P(x)−x
Hint

Turn both given values into zeros.

Answer and reasoning

P(x)−x. P(x)−x vanishes at 1 and 2. The roots are distinct, so both linear factors divide it together.

6. For P with integer coefficients, which divisor of P(8)−P(3) is guaranteed?

Stretch

  1. 8
  2. 3
  3. 11
  4. 5
Hint

Use a−b dividing P(a)−P(b).

Answer and reasoning

5. The input difference is 8−3=5, and each monomial difference is divisible by 5.

7. x−a divides P(x) exactly when what holds?

Foundation

  1. P(0)=a
  2. P(a)=a
  3. P(1)=0
  4. P(a)=0
Hint

Use the factor theorem.

Answer and reasoning

P(a)=0. The remainder upon division by x−a is P(a).

8. Find k if x−2 divides x²+kx−6.

Core

  1. −1
  2. 2
  3. −2
  4. 1
Hint

Set P(2)=0.

Answer and reasoning

1. 4+2k−6=0 gives k=1.

9. For integer-coefficient P, why does a−b divide P(a)−P(b)?

Stretch

  1. Each ak−bka^{k}-b^{k} has factor a−b
  2. All values are equal
  3. All coefficients are prime
  4. Only linear polynomials work
Hint

Factor each difference of powers.

Answer and reasoning

Each ak−bka^{k}-b^{k} has factor a−b. A sum of integer multiples of a−b is another such multiple.

Choose your next step

Continue to Polynomial roots and multiplicity. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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