Polynomials path · A01

Before this lesson: Mathematical language and sets, Powers, radicals and logarithms, Algebraic identities and factorisation

Your goal: Recognise a polynomial and explain how its degree can change under addition.

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.

Polynomial functions: the key idea

A polynomial in x is a finite sum of terms ckxkc_{k}x^{k} with nonnegative integer exponents and coefficients from a stated number system. Its degree is the largest exponent with nonzero coefficient. A nonzero constant has degree zero; we leave the zero polynomial’s degree undefined in this lesson. Adding polynomials may cancel leading terms. Multiplying two nonzero polynomials adds degrees. Evaluation substitutes an input for every x. A root a satisfies P(a)=0; the number a and the expression x−a are different mathematical objects.

A worked example

Find the degree of (x³+2x)−(x³−5).

Expand the subtraction carefully: x³+2x−x³+5=2x+5. The cubic terms cancel, so the degree is 1, not 3. Simplify before deciding the degree.

Your turn: change one thing

Find P(−2) for P(x)=x³−2x+1.

Try this on paper before opening the explanation.

Compare your reasoning

P(−2)=−8+4+1=−3. Parentheses keep the sign of the odd power clear.

Pause and check

A trap to avoid: Calling 1/x a polynomial or assigning the zero polynomial an ordinary degree.

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 if nonzero polynomials P and Q over the reals have degrees m and n, then PQ has degree m+n.

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

Let their leading terms be axmax^{m} and bxnbx^{n} with a,b nonzero. Their product contributes abxm+nabx^{m+n}, whose coefficient is nonzero. Every other product of terms has degree smaller than m+n, so none can cancel this term. Hence the product degree is exactly m+n.

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. Which expression is a polynomial in x?

Foundation

  1. x+x\sqrt{x}+x
  2. x−2+1x^{-2}+1
  3. 3x²−x+4
  4. 1/x
Hint

Polynomial exponents are nonnegative integers.

Answer and reasoning

3x²−x+4. 3x²−x+4 has exponents 2,1,0. The others contain negative or fractional powers.

2. Find the degree of 7−2x⁴+x.

Foundation

  1. 7
  2. 4
  3. 1
  4. 3
Hint

Find the largest exponent with nonzero coefficient.

Answer and reasoning

4. The x⁴ term has nonzero coefficient −2, so the degree is 4.

3. For P(x)=x²−3x+2, find P(4).

Core

  1. 2
  2. 4
  3. 8
  4. 6
Hint

Substitute 4 throughout.

Answer and reasoning

6. 16−12+2=6.

4. Find the degree of (x³+1)−(x³−x²).

Core

  1. 6
  2. 2
  3. 0
  4. 3
Hint

Simplify first.

Answer and reasoning

2. The expression equals x²+1; its degree is 2.

5. Nonzero polynomials P,Q have degrees 2 and 5. What is deg(PQ)?

Stretch

  1. It can be 0
  2. 7
  3. 3
  4. 10
Hint

Multiply their leading terms.

Answer and reasoning

7. The leading product has degree 2+5=7 and a nonzero coefficient.

6. What does P(a)=0 tell you?

Stretch

  1. Every input is a root
  2. a is a root of P
  3. P is the zero polynomial
  4. P has degree zero
Hint

One zero value does not describe every input.

Answer and reasoning

a is a root of P. By definition a is a root. A nonzero polynomial can have roots, so this does not make P identically zero.

7. What is the degree of 7x⁴−x+2?

Foundation

  1. 1
  2. 4
  3. 7
  4. 2
Hint

Use the largest exponent with nonzero coefficient.

Answer and reasoning

4. The x⁴ coefficient is 7≠0.

8. Find P(−1) for P(x)=x³+2x²−3.

Core

  1. −6
  2. −2
  3. 0
  4. 2
Hint

Substitute carefully with signs.

Answer and reasoning

−2. −1+2−3=−2.

9. Can the degree drop when two polynomials are added?

Stretch

  1. Yes, leading coefficients may cancel
  2. No, never
  3. Only for zero inputs
  4. Only for constant polynomials
Hint

Try x² and −x²+x.

Answer and reasoning

Yes, leading coefficients may cancel. Their sum is x, whose degree is below both quadratic degrees.

Choose your next step

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

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