CLEP College Algebra — Questions and Answers

Question 1: Which is the correct expansion of (x + 5)(x - 3)?

x² - 2x - 15
x² + 2x + 15
x² + 2x - 15 (Correct answer)
x² - 2x + 15

Correct answer: x² + 2x - 15

FOIL: x² - 3x + 5x - 15 = x² + 2x - 15.

Question 2: A system with exactly one solution is called:

Inconsistent
Consistent and independent (Correct answer)
Trivial
Dependent

Correct answer: Consistent and independent

A system with exactly one solution is consistent (a solution exists) and independent (distinct lines).

Question 3: An even function satisfies:

f(-x) = -f(x)
f(0) = 0
f(-x) = f(x) (Correct answer)
f(x + 1) = f(x)

Correct answer: f(-x) = f(x)

An even function satisfies f(-x) = f(x), meaning it is symmetric about the y-axis.

Question 4: Divide 6x³ by 2x.

3x³
12x⁴
4x²
3x² (Correct answer)

Correct answer: 3x²

Divide coefficients 6/2 = 3 and subtract exponents 3 - 1 = 2, giving 3x².

Question 5: What is the product of the roots of 2x² - 6x + 4 = 0?

2 (Correct answer)
3
-3
4

Correct answer: 2

By Vieta's formulas, the product of roots = c/a = 4/2 = 2.

Question 6: Solve: 2(3x + 1) = 14

x = 3
x = 2 (Correct answer)
x = 4
x = 1

Correct answer: x = 2

Distribute to get 6x + 2 = 14, subtract 2 to get 6x = 12, then divide by 6 to get x = 2.

Question 7: What is the range of f(x) = x² for all real x?

All real numbers
y ≥ 0 (Correct answer)
y > 0
y ≤ 0

Correct answer: y ≥ 0

Since x² ≥ 0 for all real x, the range is y ≥ 0.

Question 8: If f(x) = 3x - 2, what is f(4)?

6
8
12
10 (Correct answer)

Correct answer: 10

f(4) = 3(4) - 2 = 10.

Question 9: Rationalize the denominator of 1/√5.

√5/25
√5/5 (Correct answer)
1/5
5/√5

Correct answer: √5/5

Multiply by √5/√5: (1 · √5)/(√5 · √5) = √5/5.

Question 10: Which of the following is a trinomial?

x³ + 2x - 7 (Correct answer)
3x² - 1
5x
4x⁴ + x³ - x² + 1

Correct answer: x³ + 2x - 7

A trinomial has exactly three terms; x³ + 2x - 7 has three terms.

Question 11: What is the value of 2⁵?

32 (Correct answer)
10
64
25

Correct answer: 32

2⁵ means 2 multiplied by itself 5 times: 2×2×2×2×2 = 32.

Question 12: Simplify: (x³ · x⁻¹) / x²

x²
x⁴
1 (Correct answer)
x⁻²

Correct answer: 1

x³ · x⁻¹ = x², then x² / x² = x⁰ = 1.

Question 13: What is the inverse of f(x) = 2x + 6?

f⁻¹(x) = (x + 6)/2
f⁻¹(x) = (x - 6)/2 (Correct answer)
f⁻¹(x) = x/2 + 6
f⁻¹(x) = 2x - 6

Correct answer: f⁻¹(x) = (x - 6)/2

Swap x and y: x = 2y + 6, then solve for y: y = (x - 6)/2.

Question 14: Simplify: √(x⁴), assuming x ≥ 0

x
x³
x⁴
x² (Correct answer)

Correct answer: x²

√(x⁴) = x^(4/2) = x².

Question 15: Factor: x² - 9.

(x - 3)(x - 3)
(x - 9)(x + 1)
(x + 9)(x - 1)
(x + 3)(x - 3) (Correct answer)

Correct answer: (x + 3)(x - 3)

x² - 9 = x² - 3² is a difference of squares, factoring as (x + 3)(x - 3).

Question 16: Which expression is equivalent to ∛(x⁶)?

x⁹
x³
x¹⁸
x² (Correct answer)

Correct answer: x²

∛(x⁶) = x^(6/3) = x².

Question 17: If two equations are identical, the system has:

Two solutions
One solution
Infinitely many solutions (Correct answer)
No solution

Correct answer: Infinitely many solutions

Identical equations describe the same line, so every point on it is a solution.

Question 18: When a system of two linear equations has no solution, geometrically the lines are:

Intersecting at one point
Parallel (Correct answer)
The same line
Perpendicular

Correct answer: Parallel

No solution means the lines are parallel and never intersect.

Question 19: Factor completely: 3x³ - 12x.

3x(x + 2)(x - 2) (Correct answer)
x(3x² - 12)
3x(x² - 4)
3(x³ - 4x)

Correct answer: 3x(x + 2)(x - 2)

Factor out 3x to get 3x(x² - 4), then factor the difference of squares to get 3x(x + 2)(x - 2).

Question 20: Multiply: (x + 3)(x - 2).

x² - x - 6
x² + x - 6 (Correct answer)
x² + 5x - 6
x² - 5x + 6

Correct answer: x² + x - 6

Use FOIL: x² - 2x + 3x - 6 = x² + x - 6.

Question 21: The vertical line test determines whether a graph is:

An inequality
A linear equation
A function (Correct answer)
A polynomial

Correct answer: A function

The vertical line test checks if any vertical line intersects the graph more than once; if not, it is a function.

Question 22: Which of the following relations is a function?

{(1,2),(1,3),(2,4)}
{(0,0),(1,1),(1,-1)}
{(2,1),(2,3),(2,5)}
{(1,2),(2,3),(3,4)} (Correct answer)

Correct answer: {(1,2),(2,3),(3,4)}

A function requires each input to have exactly one output; only the second set has no repeated x-values.

Question 23: What is the solution to -x ≥ 3?

x ≥ 3
x ≥ -3
x ≤ -3 (Correct answer)
x ≤ 3

Correct answer: x ≤ -3

Multiply both sides by -1 and flip: x ≤ -3.

Question 24: Subtract: (4x² - 3x + 2) - (x² + 2x - 5).

5x² - 5x + 7
3x² + 5x - 7
3x² - x + 7
3x² - 5x + 7 (Correct answer)

Correct answer: 3x² - 5x + 7

Distribute the negative: 4x² - 3x + 2 - x² - 2x + 5 = 3x² - 5x + 7.

Question 25: If y = 2x + 1, what is y when x = 4?

9 (Correct answer)
8
10
7

Correct answer: 9

Substitute x = 4 to get y = 2(4) + 1 = 9.

Question 26: What are the solutions to x² - 5x + 6 = 0?

x = 1 and x = 6
x = -2 and x = -3
x = 2 and x = 3 (Correct answer)
x = -1 and x = -6

Correct answer: x = 2 and x = 3

Factor as (x - 2)(x - 3) = 0, giving x = 2 and x = 3.

Question 27: Which form of a quadratic is easiest for reading the roots directly?

Vertex form
Factored form (Correct answer)
Standard form
Point-slope form

Correct answer: Factored form

Factored form y = a(x - r₁)(x - r₂) directly shows the roots r₁ and r₂.

Question 28: What is 4^(1/2)?

16
4
2 (Correct answer)
8

Correct answer: 2

4^(1/2) equals √4 = 2, since 2² = 4.

Question 29: What is the slope of a horizontal line?

0 (Correct answer)
Undefined
1
-1

Correct answer: 0

A horizontal line has no rise for any run, so its slope is 0.

Question 30: Subtract: (5x² - 3x + 2) - (2x² + x - 4)

3x² - 4x - 2
3x² - 4x + 6 (Correct answer)
3x² + 4x - 6
7x² - 2x - 2

Correct answer: 3x² - 4x + 6

Distribute the negative: (5-2)x² = 3x², (-3-1)x = -4x, (2+4) = 6, giving 3x² - 4x + 6.

Question 31: Solve: y = 3x - 1 and y = x + 5

x = 1, y = 2
x = 3, y = 8 (Correct answer)
x = 4, y = 11
x = 2, y = 7

Correct answer: x = 3, y = 8

Set 3x - 1 = x + 5, so 2x = 6 and x = 3; then y = 8.

Question 32: Using the quadratic formula, solve x² - 2x - 3 = 0.

x = 3 and x = -1 (Correct answer)
x = -3 and x = 1
x = 1 and x = -2
x = 2 and x = -1

Correct answer: x = 3 and x = -1

Using x = (2 ± √(4 + 12))/2 = (2 ± 4)/2 gives x = 3 or x = -1.

Question 33: Multiply: (x - 4)(x + 4).

x² + 8x - 16
x² - 16 (Correct answer)
x² + 16
x² - 8x + 16

Correct answer: x² - 16

This is a difference of squares: (a - b)(a + b) = a² - b² = x² - 16.

Question 34: Factor out the GCF: 6x³ + 9x².

3x(2x² + 3x)
3x²(2x + 3) (Correct answer)
9x²(x + 1)
6x²(x + 9)

Correct answer: 3x²(2x + 3)

The GCF of 6x³ and 9x² is 3x², giving 3x²(2x + 3).

Question 35: Solve: 4x + 1 ≥ 2x + 9

x ≥ 2
x ≤ 2
x ≥ 4 (Correct answer)
x ≤ 4

Correct answer: x ≥ 4

Subtract 2x to get 2x + 1 ≥ 9, subtract 1 to get 2x ≥ 8, divide to get x ≥ 4.

Question 36: If f(x) = x² + 1 and g(x) = 2x, what is (f ∘ g)(3)?

19
13
7
37 (Correct answer)

Correct answer: 37

g(3) = 6; then f(6) = 36 + 1 = 37.

Question 37: What is √144?

14
11
12 (Correct answer)
13

Correct answer: 12

12 × 12 = 144, so √144 = 12.

Question 38: What is the end behavior of y = -x³ + 2x as x → +∞?

y → 0
y → +∞
y → 2
y → -∞ (Correct answer)

Correct answer: y → -∞

With odd degree and negative leading coefficient, y goes to -∞ as x → +∞.

Question 39: What is the domain of f(x) = 1/(x - 5)?

x ≠ 5 (Correct answer)
x > 5
All real numbers
x ≠ 0

Correct answer: x ≠ 5

The denominator cannot be zero: x - 5 ≠ 0, so x ≠ 5.

Question 40: What is the value of 8^(2/3)?

4 (Correct answer)
8
2
16

Correct answer: 4

8^(2/3) = (8^(1/3))² = 2² = 4.

Question 41: What is the result of (2x + 3)(3x - 1)?

6x² - 7x - 3
6x² + 7x - 3 (Correct answer)
6x² + 7x + 3
6x² - 7x + 3

Correct answer: 6x² + 7x - 3

FOIL: 6x² - 2x + 9x - 3 = 6x² + 7x - 3.

Question 42: Which polynomial has roots at x = 1 and x = -5?

x² - 4x - 5
x² - 4x + 5
x² + 6x + 5
x² + 4x - 5 (Correct answer)

Correct answer: x² + 4x - 5

(x - 1)(x + 5) = x² + 5x - x - 5 = x² + 4x - 5.

Question 43: Factor: x² - 2x - 8.

(x + 4)(x - 2)
(x - 4)(x + 2) (Correct answer)
(x - 4)(x - 2)
(x + 4)(x + 2)

Correct answer: (x - 4)(x + 2)

Find two numbers that multiply to -8 and add to -2: -4 and 2, giving (x - 4)(x + 2).

Question 44: If f(x) = 5 for all x, the function is called a:

Quadratic function
Constant function (Correct answer)
Step function
Linear function

Correct answer: Constant function

A function that outputs the same value for every input is a constant function.

Question 45: Which describes an increasing function on an interval?

As x decreases, y increases
As x increases, y increases (Correct answer)
As x increases, y stays constant
As x increases, y decreases

Correct answer: As x increases, y increases

An increasing function produces larger outputs as input values increase.

Question 46: What transformation does y = f(x) + 3 represent?

Shift 3 units down
Shift 3 units up (Correct answer)
Shift 3 units right
Shift 3 units left

Correct answer: Shift 3 units up

Adding a constant outside the function shifts the graph vertically upward by 3 units.

Question 47: Which graph shows x ≤ -2?

Open circle at -2, shading right
Closed circle at -2, shading left (Correct answer)
Closed circle at -2, shading right
Open circle at -2, shading left

Correct answer: Closed circle at -2, shading left

x ≤ -2 uses a closed circle at -2 with shading to the left.

Question 48: Solve by elimination: 3x + 2y = 16 and x - 2y = 0

x = 2, y = 4
x = 3, y = 3
x = 5, y = 1
x = 4, y = 2 (Correct answer)

Correct answer: x = 4, y = 2

Adding the equations gives 4x = 16, so x = 4; substituting gives y = 2.

Question 49: What is the axis of symmetry for y = x² - 6x + 8?

x = -6
x = 6
x = -3
x = 3 (Correct answer)

Correct answer: x = 3

The axis of symmetry is x = -b/(2a) = -(-6)/(2·1) = 3.

Question 50: What is the leading coefficient of -3x⁴ + 2x² - x?

1
4
3
-3 (Correct answer)

Correct answer: -3

The leading coefficient is the coefficient of the highest-degree term, which is -3.

Question 51: What is 27^(1/3)?

9
6
3 (Correct answer)
81

Correct answer: 3

27^(1/3) is the cube root of 27; since 3³ = 27, the answer is 3.

Question 52: What is the x-intercept of f(x) = 2x - 8?

x = 4 (Correct answer)
x = 2
x = -8
x = -4

Correct answer: x = 4

Set f(x) = 0: 2x - 8 = 0 gives x = 4.

Question 53: What is √72 in simplified radical form?

6√3
8√3
4√6
6√2 (Correct answer)

Correct answer: 6√2

√72 = √(36 × 2) = √36 · √2 = 6√2.

Question 54: In interval notation, how is x ≥ -1 written?

(-∞, -1]
[-1, ∞) (Correct answer)
(-1, ∞)
(-∞, -1)

Correct answer: [-1, ∞)

x ≥ -1 includes -1 (bracket) and extends to positive infinity: [-1, ∞).

Question 55: Which function graphs as a straight line?

f(x) = x²
f(x) = 2x + 1 (Correct answer)
f(x) = 1/x
f(x) = √x

Correct answer: f(x) = 2x + 1

f(x) = 2x + 1 is a linear function and graphs as a straight line.

Question 56: What transformation does y = f(x - 2) represent?

Shift 2 units right (Correct answer)
Shift 2 units up
Shift 2 units down
Shift 2 units left

Correct answer: Shift 2 units right

Replacing x with (x - 2) shifts the graph 2 units to the right.

Question 57: What is the value of 3⁻²?

-9
1/6
1/9 (Correct answer)
-1/9

Correct answer: 1/9

A negative exponent means reciprocal: 3⁻² = 1/3² = 1/9.

Question 58: Simplify: √50

5√2 (Correct answer)
5
10√2
25√2

Correct answer: 5√2

√50 = √(25 × 2) = √25 · √2 = 5√2.

Question 59: What notation means 'f composed with g'?

f ∘ g (Correct answer)
f · g
f + g
f ÷ g

Correct answer: f ∘ g

Function composition is written f ∘ g, meaning f(g(x)).

Question 60: For f(x) = |x|, what is the minimum value?

1
-1
There is no minimum
0 (Correct answer)

Correct answer: 0

Since |x| ≥ 0 for all real x, the minimum value is 0, achieved at x = 0.

Question 61: What is the domain of f(x) = √(x - 3)?

All real numbers
x ≥ 0
x > 3
x ≥ 3 (Correct answer)

Correct answer: x ≥ 3

The radicand must be non-negative: x - 3 ≥ 0, so x ≥ 3.

CLEP College Algebra

The CLEP College Algebra exam tests mastery of algebraic concepts typically covered in a one-semester college algebra course, including functions, equations, inequalities, and algebraic operations. A passing score can earn undergraduate college credit at thousands of institutions.

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90 min

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70%

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