Q. Factor the expression: 2x^2 + 8x.
A.
2x(x + 4)
B.
2(x^2 + 4x)
C.
x(2x + 8)
D.
2x^2(1 + 4)
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Solution
First, we can factor out the greatest common factor, which is 2x. This gives us 2x(x + 4).
Correct Answer:
A
— 2x(x + 4)
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Q. Factor the expression: 4x^2 - 25.
A.
(2x - 5)(2x + 5)
B.
(4x - 5)(4x + 5)
C.
(2x - 25)(2x + 25)
D.
(4x - 5)(4x + 5)
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Solution
The expression 4x^2 - 25 is a difference of squares. It can be factored as (2x - 5)(2x + 5).
Correct Answer:
A
— (2x - 5)(2x + 5)
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Q. Factor the expression: x^2 + 6x + 9.
A.
(x + 3)(x + 3)
B.
(x + 2)(x + 4)
C.
(x - 3)(x - 3)
D.
(x + 1)(x + 9)
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Solution
The expression x^2 + 6x + 9 is a perfect square trinomial. It factors to (x + 3)(x + 3) or (x + 3)^2.
Correct Answer:
A
— (x + 3)(x + 3)
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Q. Factor the expression: x^2 + 7x + 10.
A.
(x + 2)(x + 5)
B.
(x - 2)(x - 5)
C.
(x + 1)(x + 10)
D.
(x - 1)(x - 10)
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Solution
To factor x^2 + 7x + 10, we need two numbers that multiply to 10 and add to 7. The numbers 2 and 5 work. Thus, the factorization is (x + 2)(x + 5).
Correct Answer:
A
— (x + 2)(x + 5)
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Q. Factor the expression: x^2 - 5x + 6.
A.
(x - 2)(x - 3)
B.
(x + 2)(x + 3)
C.
(x - 1)(x - 6)
D.
(x + 1)(x + 6)
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Solution
To factor x^2 - 5x + 6, we look for two numbers that multiply to 6 and add to -5. The numbers -2 and -3 work. Thus, the factorization is (x - 2)(x - 3).
Correct Answer:
A
— (x - 2)(x - 3)
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Q. Factor the expression: x^2 - 9.
A.
(x - 3)(x + 3)
B.
(x - 4)(x + 4)
C.
(x - 1)(x + 1)
D.
(x + 3)(x + 3)
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Solution
The expression x^2 - 9 is a difference of squares. It can be factored as (x - 3)(x + 3).
Correct Answer:
A
— (x - 3)(x + 3)
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Q. Factor the polynomial: 3x^2 + 12x
A.
3x(x + 4)
B.
3(x^2 + 4)
C.
x(3x + 12)
D.
3x^2 + 4x
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Solution
Factor out the common term 3x: 3x(x + 4).
Correct Answer:
A
— 3x(x + 4)
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Q. Solve the equation: 5x + 3 = 2x + 12
A.
x = 3
B.
x = 4
C.
x = 5
D.
x = 6
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Solution
Subtract 2x from both sides: 3x + 3 = 12. Then subtract 3: 3x = 9. Finally, divide by 3: x = 3.
Correct Answer:
B
— x = 4
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Q. Solve the inequality: 2x - 3 < 7
A.
x < 5
B.
x > 5
C.
x < 2
D.
x > 2
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Solution
Add 3 to both sides: 2x < 10. Then divide by 2: x < 5.
Correct Answer:
A
— x < 5
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Q. Solve the inequality: 3x + 4 > 10
A.
x > 2
B.
x < 2
C.
x > 3
D.
x < 3
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Solution
Subtract 4 from both sides: 3x > 6. Then divide by 3: x > 2.
Correct Answer:
A
— x > 2
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Q. What are the roots of the equation: x^2 - 4 = 0?
A.
-2, 2
B.
0, 4
C.
1, -1
D.
2, -2
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Solution
Set the equation to zero: x^2 - 4 = 0. Factor as (x - 2)(x + 2) = 0. Thus, the roots are x = -2 and x = 2.
Correct Answer:
A
— -2, 2
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Q. What are the roots of the quadratic equation: x^2 + 6x + 9 = 0?
A.
-3, -3
B.
3, 3
C.
0, 9
D.
1, -1
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Solution
Factor the equation as (x + 3)(x + 3) = 0. Thus, the root is x = -3 (a double root).
Correct Answer:
A
— -3, -3
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Q. What is the factored form of 2x^2 - 8?
A.
2(x - 4)(x + 4)
B.
2(x - 2)(x + 2)
C.
2(x - 4)
D.
2(x + 4)
Show solution
Solution
Factor out the common term 2: 2(x^2 - 4). Then factor x^2 - 4 as (x - 2)(x + 2). Thus, the final factorization is 2(x - 4)(x + 4).
Correct Answer:
A
— 2(x - 4)(x + 4)
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Q. What is the factored form of the expression 4x^2 - 12x?
A.
4x(x - 3)
B.
2x(2x - 3)
C.
4(x^2 - 3)
D.
2(2x^2 - 6x)
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Solution
The greatest common factor is 4x. Factoring it out gives us 4x(x - 3).
Correct Answer:
A
— 4x(x - 3)
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Q. What is the solution to the inequality 3x - 7 < 2?
A.
x < 3
B.
x < 1
C.
x > 1
D.
x > 3
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Solution
To solve 3x - 7 < 2, add 7 to both sides: 3x < 9. Then divide by 3: x < 3.
Correct Answer:
B
— x < 1
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