1.
Expand and simplify: (4s + 9t)(5s - 4t - 3).
Correct answer: 20s² + 29st - 12s - 36t² - 27t. (4s + 9t)(5s - 4t - 3) = 20s² - 16st - 12s + 45st - 36t² - 27t = 20s² + 29st - 12s - 36t² - 27t.
2.
Simplify: -2(x - 3)² + 2(x + 3).
Correct answer: -2x² + 14x - 12. -2(x - 3)² + 2(x + 3) = -2(x - 3)(x - 3) + 2(x + 3) = -2(x² - 6x + 9) + 2x + 6 = -2x² + 12x - 18 + 2x + 6 = -2x² + 14x - 12.
3.
Expand and simplify: (n² - 2n + 3)(-2n² + 7n + 8).
Correct answer: -2n⁴ + 11n³ - 12n² + 5n + 24. Multiply each term: -2n⁴ + 7n³ + 8n² + 4n³ - 14n² - 16n - 6n² + 21n + 24 = -2n⁴ + 11n³ - 12n² + 5n + 24.
4.
Simplify: (a - 4)(a² + 3a - 2).
Correct answer: a³ - a² - 14a + 8. (a - 4)(a² + 3a - 2) = a³ + 3a² - 2a - 4a² - 12a + 8 = a³ - a² - 14a + 8.
5.
Expand and simplify: (h - 6)(h + 11).
Correct answer: h² + 5h - 66. (h - 6)(h + 11) = h² + 11h - 6h - 66 = h² + 5h - 66.
6.
Expand and simplify: 3(1 - 2t)(9 + 4t).
Correct answer: -24t² - 42t + 27. First, (1 - 2t)(9 + 4t) = 9 + 4t - 18t - 8t² = 9 - 14t - 8t². Then multiply by 3: 27 - 42t - 24t² = -24t² - 42t + 27.
7.
Simplify: (3x - 1)³.
Correct answer: 27x³ - 27x² + 9x - 1. (3x - 1)³ = (3x - 1)(3x - 1)(3x - 1). First, (3x - 1)(3x - 1) = 9x² - 6x + 1. Then (9x² - 6x + 1)(3x - 1) = 27x³ - 27x² + 9x - 1.
8.
Expand and simplify: (10v - 13w)(10v + 13w).
Correct answer: 100v² - 169w². This is a difference of squares: (10v - 13w)(10v + 13w) = (10v)² - (13w)² = 100v² - 169w².
9.
Expand and simplify: (3c + 2)(2c - 7) + 3(-2c + 1)(7c - 5).
Correct answer: -36c² + 34c - 29. (3c + 2)(2c - 7) = 6c² - 17c - 14. (-2c + 1)(7c - 5) = -14c² + 17c - 5. Multiply by 3: -42c² + 51c - 15. Add: 6c² - 17c - 14 - 42c² + 51c - 15 = -36c² + 34c - 29.
10.
Which multiplication sentence does this set of algebra tiles represent?
Correct answer: (2x + 2)(2x + 2). The original question uses an algebra tiles diagram. Please insert the image manually after importing this question.
11.
Harpreet is 8 years older than Jennifer. The product of their ages is 153. Which equation should be used?
Correct answer: j(j + 8) = 153. Let Jennifer's age be j. Harpreet is 8 years older, so Harpreet's age is j + 8. Their product is 153, so the equation is j(j + 8) = 153.
12.
Expand and simplify: (6x - y)(3x + 8y) - (2x - 3y)².
Correct answer: 14x² + 57xy - 17y². (6x - y)(3x + 8y) = 18x² + 45xy - 8y². (2x - 3y)² = 4x² - 12xy + 9y². Subtract: 18x² + 45xy - 8y² - 4x² + 12xy - 9y² = 14x² + 57xy - 17y².
13.
Multiply: (3a² + b³)(3a² - b³).
Correct answer: 9a⁴ - b⁶. This is a difference of squares: (3a² + b³)(3a² - b³) = (3a²)² - (b³)² = 9a⁴ - b⁶.
14.
Simplify: (3x² + yz)(3x² - yz).
Correct answer: 9x⁴ - y²z². This is a difference of squares: (3x² + yz)(3x² - yz) = (3x²)² - (yz)² = 9x⁴ - y²z².
15.
Simplify: (x + y)(x - y)(4x + y).
Correct answer: 4x³ + x²y - 4xy² - y³. First, (x + y)(x - y) = x² - y². Then (x² - y²)(4x + y) = 4x³ + x²y - 4xy² - y³.
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