Online Test — Factorization
10 Questions • 15 min • Chapter MCQ
15:00
Question 1 of 10
medium
According to the Factor Theorem, (x − a) is a factor of p(x) if:
p(a)=0
p(0)=a
p(1)=a
a=0
Explanation: Factor Theorem: p(a)=0 ⇔ (x−a) is factor
Question 2 of 10
medium
Find the remainder when \(x^{3} - 2x^{2} + 4x - 5\) is divided by (x − 1).
-2
-1
0
1
Explanation: p(1)=1−2+4−5=−2
Question 3 of 10
medium
Which of the following is a factor of \(x^{3} - 4x^{2} + x + 6\)?
(x−1)
(x−2)
(x−3)
(x+2)
Explanation: Try x=−2: p(−2)=−8−16−2+6=−20 ≠0? Wait p(−2)=-8-16-2+6=-20. Try x=2: 8-16+2+6=0→(x-2) is factor. But (x+2) not factor. Check options: B (x−2). Yes answer B. Let me correct: p(2)=0, so (x−2) is factor.
Question 4 of 10
medium
For what value of k is (x − 1) a factor of \(x^{3} - 2x^{2} + kx + 4\)?
-3
-2
2
3
Explanation: p(1)=1−2+k+4=3+k=0→k=−3
Question 5 of 10
medium
The remainder when \(2x^{3} - 3x^{2} + 4x - 1\) is divided by (2x − 1) is:
0
1/4
-1/4
1/2
Explanation: 2x-1=0→x=1/2; p(1/2)=2(1/8)-3(1/4)+4(1/2)-1=0.25-0.75+2-1=0.5? 0.25-0.75=-0.5, -0.5+2=1.5, 1.5-1=0.5=1/2. That's D. Wait recalc: 2/8=0.25, -3/4=-0.75, 4/2=2, -1. Sum:0.25-0.75=-0.5, -0.5+2=1.5, 1.5-1=0.5=1/2. So D.
Question 6 of 10
medium
Factorize: \(x^{3} - 6x^{2} + 11x - 6\)
(x−1)(x−2)(x+3)
(x−1)(x−2)(x−3)
(x+1)(x−2)(x−3)
(x+1)(x+2)(x+3)
Explanation: Standard factorization: roots 1,2,3
Question 7 of 10
medium
If p(x) is divided by (x + 3), the remainder is p(___):
3
-3
0
1
Explanation: For (x + a), remainder = p(−a), so for (x+3), remainder = p(−3)
Question 8 of 10
medium
One factor of \(2x^{3} - 3x^{2} - 3x + 2\) is:
(x+1)
(x−2)
(x−1)
(x+2)
Explanation: Try x=−2: −16−12+6+2=−20≠0? Try x=2:16−12−6+2=0→(x−2) factor. So B. Correct: p(2)=16-12-6+2=0, so (x−2).
Question 9 of 10
medium
The remainder when \(x^{4} + x^{3} - 2x^{2} + x + 1\) is divided by (x − 1) is:
0
1
2
3
Explanation: p(1)=1+1−2+1+1=2
Question 10 of 10
medium
Factorize: \(x^{3} - 23x^{2} + 142x - 120\). One factor is (x − 1). The other factors are:
(x−10)(x−12)
(x−8)(x−15)
(x−9)(x−14)
(x−11)(x−13)
Explanation: Divide by (x−1): \(x^{2}-22x+120\) = (x−10)(x−12)