Write polynomials of degrees 1, 2 and 3.
Answer: For example: degree 1: 3x + 5; degree 2: x2 − 4x + 1; degree 3: 2x3 + x − 7. (Many answers are correct.)
Step-by-step solution
Idea: To get degree n, make sure the highest power of the variable is exactly n, with a non-zero coefficient. The other terms can be anything of lower power.
- Degree 1 (linear): the highest power must be 1. Example: 3x + 5.½ mark
- Degree 2 (quadratic): the highest power must be 2. Example: x2 − 4x + 1.½ mark
- Degree 3 (cubic): the highest power must be 3. Example: 2x3 + x − 7. (It does not need an x2 term.)½ mark
- Each example uses one variable, only whole-number powers, and its highest-power term has a non-zero coefficient, so each is a polynomial of the stated degree. Other correct answers: 5 − 2y; 7t2; z3 − 1.½ mark
Degree 1: 3x + 5; degree 2: x2 − 4x + 1; degree 3: 2x3 + x − 7 (any polynomials whose highest powers are 1, 2 and 3 are correct).
Answer to write in the exam
Degree 1: 3x + 5
Degree 2: x2 − 4x + 1
Degree 3: 2x3 + x − 7
Common mistakes that cost marks
- Writing 0x2 + 3x + 1 as a degree 2 polynomial. A zero coefficient means the term is not really there; this is degree 1.
- Using a power of the variable in a denominator, such as 1x + 2. That is not a polynomial, because powers must be whole numbers 0, 1, 2, …
- Thinking a cubic must have all four terms x3, x2, x and a constant. Only the x3 term is needed.
How this can come in the exam
MCQ (1 mark)
Which of the following is a polynomial of degree 2?
- 5y − 2
- 4 − y2 + y
- y3 − y2
- 0y2 + 7y
Show answer
(B) 4 − y2 + y
4 − y2 + y has highest power 2. Option (D) is really 7y, degree 1.
Short answer (2 marks)
Write a cubic polynomial in t with exactly two terms, and a linear polynomial in t whose constant term is −4.
Show answer
Cubic with two terms, for example 3t3 + 8 (1 mark). Linear with constant −4, for example 2t − 4 (1 mark). Any correct examples score.Try one yourself
Write a polynomial of degree 4 in the variable p that has exactly three terms.
Show answer
For example p4 − 3p + 2 (any three-term polynomial whose highest power of p is 4).
More questions like this
- What are the coefficients of x2 and x3 in the polynomial x4 − 3x3 + 6x2 − 2x + 7?
- What is the coefficient of z in the polynomial 4z3 + 5z2 − 11?
- What is the constant term of the polynomial 9x3 + 5x2 − 8x −10?
- The perimeter of a square of side x is 4x, which is a linear polynomial in the variable x.
- Find the perimeter of squares with sides 1 cm, 1.5 cm, 2 cm, 2.5 cm and 3 cm. What will happen to the perimeters if the sides increase by 0.5 cm?