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Degree of a polynomial · 2 marks

Find the degrees of the following polynomials:

  1. (i) 2x2 − 5x + 3
  2. (ii) y3 + 2y − 1
  3. (iii) −9
  4. (iv) 4z − 3
Answer: (i) 2 (ii) 3 (iii) 0 (iv) 1

Step-by-step solution

Idea: The degree of a polynomial in one variable is the highest power of the variable that appears (with a non-zero coefficient). A constant like −9 can be written −9x0, so its degree is 0.

(i) 2x2 − 5x + 3

  1. Powers of x present: x2, x1 (in −5x) and x0 (in the constant 3). The highest is 2.½ mark
Degree 2 (a quadratic polynomial).

(ii) y3 + 2y − 1

  1. Powers of y present: 3, 1 and 0. The highest is 3. (There is no y2 term, but that does not matter: we only need the highest power.)½ mark
Degree 3 (a cubic polynomial).

(iii) −9

  1. −9 has no variable. It can be written as −9x0, because x0 = 1. So the highest power is 0.½ mark
Degree 0 (a constant polynomial).

(iv) 4z − 3

  1. 4z = 4z1, and the constant −3 = −3z0. The highest power of z is 1.½ mark
Degree 1 (a linear polynomial).
(i) 2 (ii) 3 (iii) 0 (iv) 1

Answer to write in the exam

(i)

Highest power of x = 2

∴ Degree = 2

(ii)

Highest power of y = 3

∴ Degree = 3

(iii)

−9 = −9x0

∴ Degree = 0

(iv)

Highest power of z = 1

∴ Degree = 1

Common mistakes that cost marks

  • Giving the degree of −9 as 1 or saying it has no degree. A non-zero constant has degree 0.
  • Adding the powers (for example 3 + 1 = 4 for y3 + 2y − 1). The degree is the highest power, not the sum.
  • Reading the coefficient as the degree, for example saying 4z − 3 has degree 4. The 4 is a coefficient; the power of z is 1.

How this can come in the exam

MCQ (1 mark)

The degree of the polynomial 7 − 3p + 5p4 − p2 is

  1. 7
  2. 2
  3. 4
  4. 5
Show answer

(C) 4
The highest power of p is 4 (in 5p4).

Assertion–Reason (1 mark)

Assertion (A): The polynomial 12 has degree 0.
Reason (R): A non-zero constant c can be written as cx0.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
12 = 12x0, so the highest power of x is 0. R is the reason A is true.

Try one yourself

Find the degrees of (a) 6t2 + t5 − 1 (b) 8 − 2m (c) 15.

Show answer

(a) 5 (b) 1 (c) 0.

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