Vidaara.orgClass 10 · Mathematics
CodeVID-M10-14-FAC-01
Factor Theorem — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
$(x-a)$ is a factor of $p(x)$ iff:
- A.$p(a)=1$
- B.$p(a)=0$
- C.$p(0)=0$
- D.$p(a)>0$
2.
If $p(2)=0$, a factor is:
- A.$(x+2)$
- B.$(x-2)$
- C.$2x$
- D.$(x-1)$
3.
The factor theorem is a special case of the:
- A.remainder theorem
- B.binomial theorem
- C.mean value theorem
- D.Pythagoras
4.
If $(x+1)$ is a factor, then $p(-1)=$
- A.$1$
- B.$0$
- C.$-1$
- D.$2$
5.
Is $(x-1)$ a factor of $x^2-1$?
- A.No
- B.Yes
- C.only if $x>0$
- D.cannot say
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Is $(x-1)$ a factor of $x^2-3x+2$?
7.
Is $(x-2)$ a factor of $x^2-4$?
8.
Find $p(3)$ for $p(x)=x^2-9$ and state whether $(x-3)$ is a factor.
9.
Is $(x+2)$ a factor of $x^2+x-2$?
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Show that $(x-2)$ is a factor of $x^3-3x^2+4x-4$.
11.
Find $k$ if $(x-1)$ is a factor of $x^2+kx+2$.
12.
Is $(x+3)$ a factor of $x^3+27$?
13.
Find $a$ if $(x+1)$ is a factor of $x^3+3x^2+ax+5$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Using the factor theorem, show that $(x-1),(x+2)$ and $(x-3)$ are factors of $x^3-2x^2-5x+6$.
15.
Find $a$ and $b$ if $(x-1)$ and $(x+1)$ are factors of $x^3+ax^2+bx-1$.
Answer Key
Section A — Multiple Choice Questions
- (B) $p(a)=0$
- (B) $(x-2)$
- (A) remainder theorem
- (B) $0$
- (B) Yes
Section B — Short Answer (2 marks)
- Yes ($p(1)=0$).
- Yes.
- $p(3)=0$; yes, $(x-3)$ is a factor.
- Yes.
Section C — Short Answer (3 marks)
- Yes ($p(2)=0$).
- $k=-3$.
- Yes ($p(-3)=0$).
- $a=7$.
Section D — Long Answer (5 marks)
- All three are factors (each gives $p=0$).
- $a=1,\ b=-1$.
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