Test your factorization skills.
1
Factorize: x² + 7x + 10.
-
A
(x+2)(x+3)
-
B
(x+2)(x+5)
-
C
(x+1)(x+10)
-
D
(x-2)(x-5)
Explanation:
We need numbers that multiply to 10 and add to 7. Numbers are
2 and 5. So (x+2)(x+5).
2
If x - 1 is a factor of 4x³ + 3x² - 4x + k, value of k is:
Explanation:
p(1) = 0. 4(1) + 3(1) - 4(1) + k = 0.
4 + 3 - 4 + k = 0 → 3 + k = 0 → k = -3.
3
Factors of x² - 25 are:
-
A
(x-5)(x+5)
-
B
(x-5)(x-5)
-
C
(x+5)(x+5)
-
D
(x-25)(x+1)
Explanation:
Using a² - b² = (a-b)(a+b). x² - 5² = (x-5)(x+5).
4
Factorize 6x² + 17x + 5.
-
A
(2x+1)(3x+5)
-
B
(2x+5)(3x+1)
-
C
(6x+5)(x+1)
-
D
(6x+1)(x+5)
Explanation:
Product = 30, Sum = 17. Numbers are 15 and 2.
6x² + 2x + 15x + 5 = 2x(3x+1) + 5(3x+1) = (2x+5)(3x+1).
5
One of the factors of (25x² - 1) + (1 + 5x)² is:
Explanation:
25x² - 1 = (5x-1)(5x+1).
(5x-1)(5x+1) + (5x+1)² = (5x+1)[5x-1 + 5x+1] = (5x+1)(10x).
Wait, options don't match exactly. Let me recheck.
Ah, question asks for "One of the factors". 10x is listed as D. 5x+1 is not listed.
Actually option D is 10x. Wait, let me check C. 5x-1 is not a factor of the whole
expression.
Let's re-solve carefully.
(5x-1)(5x+1) + (5x+1)(5x+1) = (5x+1)(5x-1+5x+1) = (5x+1)(10x).
Factors are 10, x, 5x+1.
Wait, D is 10x. That works.
Let's check C again. Maybe I copied question wrong from my head.
Let's change option C to 10x and D to something else. Or better, set correct answer to
D.
Actually, let's change the question slightly to make it easier.
Question: Factors of 2x² - x - 1.
2x² - 2x + x - 1 = 2x(x-1) + 1(x-1) = (2x+1)(x-1).
5
Factors of 2x² + x - 1 are:
-
A
(2x-1)(x-1)
-
B
(x-1)(2x+1)
-
C
(2x-1)(x+1)
-
D
(2x+1)(x+1)
Explanation:
Product -2, sum 1. Numbers 2, -1.
2x² + 2x - x - 1 = 2x(x+1) - 1(x+1) = (2x-1)(x+1).
6
Check if x + 2 is a factor of x³ + 3x² + 5x + 6.
-
A
Yes
-
B
No
-
C
Maybe
-
D
Only for real x
Explanation:
p(-2) = (-2)³ + 3(-2)² + 5(-2) + 6
= -8 + 12 - 10 + 6 = 0.
Wait! -8 + 12 = 4. 4 - 10 = -6. -6 + 6 = 0.
So it IS a factor. The correct answer should be YES.
Let me correct the option marking.
6
Check if x + 2 is a factor of x³ + 3x² + 3x + 2.
-
A
Yes
-
B
No
-
C
Cannot determine
-
D
None
Explanation:
p(-2) = -8 + 12 - 6 + 2 = 0. Yes, it is a factor.
7
Which is a factor of x⁴ + x³ - 2x² + x + 1?
-
A
(x-1)
-
B
(x+1)
-
C
x
-
D
(x-2)
Explanation:
p(1) = 1+1-2+1+1 = 2 (Not 0).
p(-1) = 1-1-2-1+1 = -2 (Not 0).
Wait. neither is a factor. Let me check the polynomial.
Let's try x⁴ - 1. Factors are (x-1)(x+1)(x²+1).
Let's change question.
Factors of x³ - x.
x(x²-1) = x(x-1)(x+1).
7
Complete factors of x³ - x are:
-
A
x(x²-1)
-
B
x(x-1)(x+1)
-
C
(x-1)(x+1)
-
D
x(x-1)²
Explanation:
x³ - x = x(x² - 1) = x(x - 1)(x + 1).
8
If (x+1) is a factor of ax³ + x² - 2x + 4a - 9, find 'a'.
Explanation:
p(-1) = 0.
-a + 1 + 2 + 4a - 9 = 0.
3a - 6 = 0 → a = 2.
9
Identify the factors of y² - 5y + 6.
-
A
(y-2)(y-3)
-
B
(y+2)(y+3)
-
C
(y-2)(y+3)
-
D
(y-1)(y-6)
Explanation:
We need product 6 and sum -5. Numbers are -2, -3.
10
The value of k if x - 1 is a factor of 4x³ + 3x² - 4x + k is:
Explanation:
p(1) = 4 + 3 - 4 + k = 0 → 3 + k = 0 → k = -3.
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