MA 25B — Practice Exam 1 (Review Mode)

Same questions and choices as the original PDF. Each item shows the correct choice and a full, notebook‑style explanation.

1) Identify the polynomial as a monomial, binomial, trinomial, or none of these. Give its degree: (1/5) r³ + (2/5) r²
  1. A) Trinomial, degree 2
  2. B) Binomial, degree 3 ✔
  3. C) Binomial, degree 1
  4. D) Binomial, degree 5
Why B?
  1. There are two terms → binomial.
  2. Highest exponent on r is 3 → degree 3.
2) Find the value of -5x³ - 5x² + 18 when x = 2.
  1. A) −32
  2. B) −42 ✔
  3. C) −52
  4. D) −54
Steps
  1. -5(2)³ - 5(2)² + 18 = -5(8) - 5(4) + 18
  2. = -40 - 20 + 18 = -42
3) Subtract: (7n⁵ − 14n³ + 5) − (2n³ + 2n⁵ + 12)
  1. A) 5n⁵ − 12n³ + 17
  2. B) 5n⁵ − 16n³ − 7 ✔
  3. C) −18n⁸
  4. D) 5n⁵ − 16n³ + 17
Steps
  1. Distribute the minus: 7n⁵ − 14n³ + 5 − 2n³ − 2n⁵ − 12
  2. Combine like terms: (7−2)n⁵ + (−14−2)n³ + (5−12) → 5n⁵ − 16n³ − 7.
4) Find the product using FOIL: (5x + 7)(x − 3)
  1. A) 5x² − 8x − 21 ✔
  2. B) 5x² − 21x − 8
  3. C) 5x² − 10x − 21
  4. D) 5x² − 8x − 8
Steps
  1. FOIL: 5x·x + 5x·(−3) + 7·x + 7·(−3)
  2. = 5x² − 15x + 7x − 21 = 5x² − 8x − 21.
5) Perform the division (positive exponents in final answer):
(-30x⁵ + 15x⁴ − 40x³) / (−5x⁴)
  1. A) 6x + 15x⁴ + 8/x
  2. B) 6x − 3
  3. C) 14x − 3
  4. D) 6x − 3 + 8/x ✔
Steps
  1. Divide term‑by‑term: (-30x⁵)/(−5x⁴)=6x, (15x⁴)/(−5x⁴)=−3, (−40x³)/(−5x⁴)=8/x.
  2. Answer: 6x − 3 + 8/x.
6) Perform the division: (p² + 2p − 33) ÷ (p + 7)
  1. A) p − 5
  2. B) (p + 5) + 2/(p + 7)
  3. C) (p − 2) + 5/(p + 7)
  4. D) (p − 5) + 2/(p + 7) ✔
Steps (synthetic/long division)
  1. Divide: p² ÷ p = p. Multiply back: p(p+7)=p²+7p. Subtract → -5p.
  2. Bring down -33. Divide: -5p ÷ p = -5. Multiply back: -5(p+7)=-5p-35. Subtract → remainder 2.
  3. So p − 5 + 2/(p+7).
7) Greatest common factor of the numbers 42 and 48.
  1. A) 90
  2. B) 6 ✔
  3. C) 7
  4. D) 3
Steps

Prime factors: 42=2·3·7, 48=2⁴·3. Common = 2·3=6.

8) Greatest common factor of the terms 15m⁵, 135m⁷, 225m⁶.
  1. A) 15m⁵ ✔
  2. B) 2,025m²
  3. C) 135m⁵
  4. D) 15m²
Steps

GCF of coefficients is 15; smallest power of m is 5 → 15m⁵.

9) Complete the factoring: 5x² − 45x = 5x( … )
  1. A) 9 − x
  2. B) 9 − x²
  3. C) x − 9 ✔
  4. D) x² − 9
Steps

Factor out 5x: 5x(x − 9).

10) Factor out the GCF: 72x⁹y⁶ − 120x²y⁴ − 60x⁷y².
  1. A) 12x²(6x⁷y⁶ − 10y⁴ − 5x⁵y²)
  2. B) 12x²y²(6x⁷y⁴ − 10y² − 5x⁵) ✔
  3. C) No common factor (except 1)
  4. D) 12(6x⁹y⁶ − 10x²y⁴ − 5x⁷y²)
Steps

Common to all terms: coefficient 12, x², and y². Divide each term to get 12x²y²(6x⁷y⁴ − 10y² − 5x⁵).

11) Factor by grouping: 6a³ − 8a²b + 15ab² − 20b³
  1. A) (2a² − 5b²)(3a + 4b)
  2. B) (6a² + 5b²)(a − 4b)
  3. C) (2a² + 5b)(3a − 4b)
  4. D) (2a² + 5b²)(3a − 4b) ✔
Steps
  1. Group: (6a³ − 8a²b) + (15ab² − 20b³).
  2. Factor each: 2a²(3a − 4b) + 5b²(3a − 4b).
  3. Common binomial: (3a − 4b)(2a² + 5b²).
12) Factor completely: x² − 6x − 40
  1. A) (x − 4)(x + 10)
  2. B) (x − 4)(x + 1)
  3. C) (x + 4)(x − 10) ✔
  4. D) Prime
Steps

Find numbers with product −40 and sum −6 → −10 and +4. So (x − 10)(x + 4).

13) Factor completely: 4x⁴ + 28x³ + 48x²
  1. A) x²(x + 4)(4x + 12)
  2. B) 4x²(x + 4)(x + 3) ✔
  3. C) 4²(x² + 7x + 12)
  4. D) x²(4x + 16)(x + 3)
Steps
  1. GCF: 4x² → inside x² + 7x + 12.
  2. Factor trinomial: x² + 7x + 12 = (x + 3)(x + 4).
14) Factor by grouping: 18x² − 12xy − 15xy + 10y²
  1. A) (6x − 5)(3x − 2)
  2. B) (18x − 5y)(x − 2y)
  3. C) (6x + 5y)(3x − 2y)
  4. D) (6x − 5y)(3x − 2y) ✔
Steps
  1. Group: (18x² − 12xy) + (−15xy + 10y²).
  2. Factor: 6x(3x − 2y) − 5y(3x − 2y).
  3. Common binomial: (3x − 2y)(6x − 5y).
15) Factor completely: x² + 12xy + 36y²
  1. A) (x − 6y)²
  2. B) (x + 6y)² ✔
  3. C) (x + 6y)(x − 6y)
  4. D) Prime
Steps

Perfect‑square trinomial: x² + 2·x·6y + (6y)² = (x + 6y)².

16) Factor: 128k³m − 54m⁴
  1. A) 2m(4k + 3m²)(16k² − 12km + 9km²)
  2. B) 2m(64k − 3m)(k² + 12km + 9m²)
  3. C) (8km − 6m²)(16k² + 9m²)
  4. D) 2m(4k − 3m)(16k² + 12km + 9m²) ✔
Steps
  1. GCF: 2m → 2m(64k³ − 27m³).
  2. Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²) with a=4k, b=3m.
  3. So 2m(4k − 3m)(16k² + 12km + 9m²).
17) Solve: (4y + 5)(5y + 12) = 0
  1. A) {1, 7}
  2. B) −4, −5/12
  3. C) −5/4, −12/5 ✔
  4. D) 5/4, 12/5
Steps

Zero‑product: set each factor to zero → 4y+5=0 ⇒ y=−5/4, 5y+12=0 ⇒ y=−12/5.

18) Solve: x² + 4x − 21 = 0
  1. A) {7, −3}
  2. B) {7, 3}
  3. C) {−7, 1}
  4. D) {−7, 3} ✔
Steps

Factor: (x+7)(x−3)=0 → x=−7 or x=3.

19) A rectangle has length x+5, width x−5, and area 56. Find length and width.
  1. A) width = 2; length = 28
  2. B) width = 4; length = 14 ✔
  3. C) width = 7; length = 8
  4. D) width = 1; length = 56
Steps
  1. Area: (x+5)(x−5)=56 ⇒ x²−25=56 ⇒ x²=81.
  2. x=±9; take positive dimensions: width x−5=4, length x+5=14.
20) Find three consecutive odd integers whose sum is 36 less than the product of the smaller two.
  1. A) −6, −4, −2
  2. B) 7, 9, 11 ✔
  3. C) 5, 7, 9
  4. D) 7, 9, 11, or −6, −4, −2
Steps
  1. Let integers be n, n+2, n+4. Equation: n+(n+2)+(n+4) = n(n+2) − 36.
  2. Simplify: 3n+6 = n²+2n − 36 ⇒ n² − n − 42 = 0.
  3. Factor: (n−7)(n+6)=0 ⇒ n=7 or n=−6.
  4. Both triplets work; the provided key selects the positive set 7,9,11.