MAT 1100 College Algebra Exam | Complete Practice Questions, Correct Answers & Detailed Solutions

MAT 1100 College Algebra Exam | Complete Practice
Questions, Correct Answers & Detailed Solutions


Question 1
𝑥 −3 𝑦 4
What is the simplified form of ?
𝑥 −5 𝑦 2

• A. 𝑥 2 𝑦 2
• B. 𝑥 −8 𝑦 6
• C. 𝑥 8 𝑦 −2
𝑦2
• D.
𝑥8

• Correct Answer: A
𝑥𝑎
• Detailed Rationale: Use the quotient rule for exponents ( =
𝑥𝑏
𝑥 𝑎−𝑏 ): 𝑥 −3−(−5) 𝑦 4−2 = 𝑥 −3+5 𝑦 2 = 𝑥 2 𝑦 2 .
Question 2

Simplify the radical expression √75𝑥 3 𝑦 4 assuming 𝑥 ≥ 0and 𝑦 ≥ 0.

• A. 5𝑥𝑦 2 √3𝑥

• B. 25𝑥𝑦 2 √3𝑥

• C. 5𝑥 2 𝑦 2 √3

• D. 15𝑥𝑦√5𝑥
• Correct Answer: A

, • Detailed Rationale: Factor out the largest perfect squares:
√25 ⋅ 3 ⋅ 𝑥 2 ⋅ 𝑥 ⋅ (𝑦 2 )2 = √25𝑥 2 𝑦 4 ⋅ √3𝑥 = 5𝑥𝑦 2 √3𝑥.
Question 3
6
Rationalize the denominator of .
3−√5

9+3√5
• A.
2

3+√5
• B.
2

18+6√5
• C.
4

9−3√5
• D.
2

• Correct Answer: A
• Detailed Rationale: Multiply numerator and denominator by the
6(3+√5) 18+6√5 18+6√5 9+3√5
conjugate (3 + √5): = = = .
(3−√5)(3+√5) 9−5 4 2

Question 4
Expand and simplify the expression (2𝑥 − 3)(𝑥 2 + 4𝑥 − 5).
• A. 2𝑥 3 + 5𝑥 2 − 22𝑥 + 15
• B. 2𝑥 3 + 8𝑥 2 − 10𝑥 + 15
• C. 2𝑥 3 − 3𝑥 2 − 22𝑥 − 15
• D. 2𝑥 3 + 5𝑥 2 + 22𝑥 − 15
• Correct Answer: A

, • Detailed Rationale: Distribute each term: 2𝑥 (𝑥 2 + 4𝑥 − 5) −
3(𝑥 2 + 4𝑥 − 5) = 2𝑥 3 + 8𝑥 2 − 10𝑥 − 3𝑥 2 − 12𝑥 + 15 = 2𝑥 3 +
5𝑥 2 − 22𝑥 + 15.
Question 5
Factor the polynomial completely by grouping: 2𝑥 3 − 4𝑥 2 + 3𝑥 − 6.
• A. (2𝑥 2 + 3)(𝑥 − 2)
• B. (2𝑥 2 − 3)(𝑥 + 2)
• C. (2𝑥 − 3)(𝑥 2 + 2)
• D. (2𝑥 + 3)(𝑥 − 2)
• Correct Answer: A
• Detailed Rationale: Group terms: (2𝑥 3 − 4𝑥 2 ) + (3𝑥 − 6) =
2𝑥 2 (𝑥 − 2) + 3(𝑥 − 2). Factor out the common binomial
(𝑥 − 2)to get (2𝑥 2 + 3)(𝑥 − 2).
Question 6
Factor completely: 16𝑥 4 − 81.
• A. (2𝑥 − 3)2 (2𝑥 + 3)2
• B. (4𝑥 2 − 9)(4𝑥 2 + 9)
• C. (2𝑥 − 3)(2𝑥 + 3)(4𝑥 2 + 9)
• D. (4𝑥 − 9)(4𝑥 + 9)(𝑥 2 + 1)
• Correct Answer: C
• Detailed Rationale: Use the difference of squares repeatedly:
16𝑥 4 − 81 = (4𝑥 2 − 9)(4𝑥 2 + 9) = (2𝑥 − 3)(2𝑥 + 3)(4𝑥 2 +
9).

, Question 7
Factor the sum of cubes: 8𝑥 3 + 27.
• A. (2𝑥 + 3)(4𝑥 2 − 6𝑥 + 9)
• B. (2𝑥 − 3)(4𝑥 2 + 6𝑥 + 9)
• C. (2𝑥 + 3)(4𝑥 2 + 6𝑥 + 9)
• D. (2𝑥 + 3)3
• Correct Answer: A
• Detailed Rationale: Use the sum of cubes formula 𝑎3 + 𝑏 3 =
(𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 )where 𝑎 = 2𝑥and 𝑏 = 3: (2𝑥 +
3)((2𝑥 )2 − (2𝑥 )(3) + 32 ) = (2𝑥 + 3)(4𝑥 2 − 6𝑥 + 9).
Question 8
𝑥 2 −9
Simplify the rational expression .
𝑥 2 +5𝑥+6
𝑥−3
• A.
𝑥+2
𝑥+3
• B.
𝑥−2
𝑥−3
• C.
𝑥−2
𝑥+3
• D.
𝑥+2

• Correct Answer: A
• Detailed Rationale: Factor numerator and denominator:
(𝑥−3)(𝑥+3) 𝑥−3
(𝑥+2)(𝑥+3)
. Cancel out the common factor (𝑥 + 3)to obtain .
𝑥+2

Question 9