What are the coefficients of x2 and x3 in the polynomial x4 − 3x3 + 6x2 − 2x + 7?
Answer: Coefficient of x2 = 6; coefficient of x3 = −3.
Step-by-step solution
Idea: The coefficient of xn is the number (with its sign) that multiplies xn. Rewrite every “−” as “+ (−)” so the sign stays with its term.
- Write the polynomial as a sum of terms: x4 + (−3)x3 + 6x2 + (−2)x + 7.
- The number multiplying x2 is 6, so the coefficient of x2 is 6.½ mark
- The number multiplying x3 is −3, so the coefficient of x3 is −3 (the minus sign belongs to it).½ mark
Coefficient of x2 is 6 and coefficient of x3 is −3.
Answer to write in the exam
x4 − 3x3 + 6x2 − 2x + 7
∴ Coefficient of x2 = 6
∴ Coefficient of x3 = −3
Common mistakes that cost marks
- Writing the coefficient of x3 as 3. The term is −3x3, so the coefficient is −3.
- Giving the term (6x2) instead of the coefficient (6).
- Mixing up the powers and reading the coefficient of x3 from the x4 term (1).
How this can come in the exam
MCQ (1 mark)
The coefficient of y2 in 2y3 − 9y2 + y − 4 is
- 9
- −9
- 2
- 1
Show answer
(B) −9
The term in y2 is −9y2, so the coefficient is −9.
Assertion–Reason (1 mark)
Assertion (A): In x3 − x + 5, the coefficient of x is −1.
Reason (R): −x means (−1) × x.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
−x = (−1)x, so its coefficient is −1; R explains A.
Try one yourself
Write the coefficients of x4, x2 and x in 5x4 − x2 + 8x − 1.
Show answer
x4: 5; x2: −1; x: 8.
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