Write a polynomial of degree 3 in the variable x, in which the coefficient of the x2 term is −7.
Answer: For example x3 − 7x2 + 2x + 1 (any polynomial whose highest power is x3 and whose x2 term is −7x2).
Step-by-step solution
Idea: Degree 3 means the highest power of x is 3 (with a non-zero coefficient). Then put −7 in front of x2; the other terms are free.
- Start with an x3 term (non-zero coefficient), for example x3, so that the degree is 3.½ mark
- Add the term −7x2, and any lower terms: x3 − 7x2 + 2x + 1. Other correct answers: 4x3 − 7x2, or −x3 − 7x2 + 5.½ mark
x3 − 7x2 + 2x + 1 (one possible answer).
Answer to write in the exam
p(x) = x3 − 7x2 + 2x + 1
Highest power of x = 3; coefficient of x2 = −7
∴ x3 − 7x2 + 2x + 1 is a required polynomial
Common mistakes that cost marks
- Writing −7x3 + x2: that puts −7 on the wrong term.
- Writing only −7x2 + 3x, which has degree 2, not 3.
- Writing 7x2 and forgetting the minus sign of the coefficient.
How this can come in the exam
MCQ (1 mark)
Which polynomial has degree 3 and coefficient of x equal to 4?
- 4x3 + x
- x3 + 4x − 2
- x2 + 4x
- 4x4 + 4x
Show answer
(B) x3 + 4x − 2
Highest power 3 and the x term is 4x.
Short answer (2 marks)
Write a polynomial of degree 2 in y whose constant term is −5 and whose coefficient of y is 0.
Show answer
For example 3y2 − 5 (2 marks): degree 2, no y term, constant −5.Try one yourself
Write a cubic polynomial in t in which the coefficient of t is −2 and the constant term is 9.
Show answer
For example t3 − 2t + 9.
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