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Question 1. What is the base in the expression (7^4)? A) 4 B) 7 C) 11 D) 28 Answer: B Explanation: In an exponential expression (a^n), the base is the number being multiplied repeatedly, here it is 7. Question 2. Which rule correctly simplifies ((3^2)^5)? A) (3^{7}) B) (3^{10}) C) (3^{5}) D) (15^{2}) Answer: B Explanation: Power‑of‑a‑power rule multiplies exponents: ((a^m)^n = a^{mn}); (2·5 = 10). Question 3. If (a^{-3}= \frac{1}{8}), what is the value of (a)? A) (- 2 ) B) (2) C) (-\frac{1}{2}) D) (\frac{1}{2}) Answer: B Explanation: (a^{-3}=1/a^{3}=1/8) ⇒ (a^{3}=8) ⇒ (a=2). Question 4. Which expression is equivalent to (\frac{x^5}{x^2})? A) (x^{7}) B) (x^{3}) C) (x^{-3}) D) (\frac{1}{x^{3}}) Answer: B Explanation: Quotient rule subtracts exponents: (5-2=3). Question 5. What is the degree of the polynomial (4x^6 - 2x^3 + 7)? A) 6 B) 4 C) 3 D) 7 Answer: A
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Explanation: Degree is the highest exponent of the variable, here 6. Question 6. Identify the term that is NOT a monomial. A) (5x^2) B) (- 3 ) C) (\frac{2}{x}) D) (9y^0) Answer: C Explanation: A monomial cannot have variables in the denominator; (\frac{2}{x}=2x^{-1}) is not a monomial under standard definition. Question 7. Which of the following is the correct product of ((x+4)(x-4))? A) (x^2 - 8x + 16) B) (x^2 - 16 ) C) (x^2 + 16) D) (x^2 + 8x + 16) Answer: B Explanation: Difference of squares: ((a+b)(a-b)=a^2-b^2); here (a=x, b=4). Question 8. Simplify ((2x^3y)^2). A) (4x^6y^2) B) (2x^6y) C) (8x^5y^2) D) (4x^5y) Answer: A Explanation: Power of a product distributes: ((ab)^n = a^n b^n); square each factor. Question 9. What is the result of ((\frac{3}{5})^{-2})? A) (\frac{9}{25}) B) (\frac{25}{9}) C) (-\frac{9}{25}) D) (-\frac{25}{9}) Answer: B Explanation: Negative exponent inverts the base: ((a/b)^{-n} = (b/a)^{n}). Question 10. Which polynomial is in standard form?
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Question 15. Simplify ( (x^2 y^3)^0). A) (0) B) (1) C) (x^2 y^3) D) Undefined Answer: B Explanation: Any non‑zero expression raised to the zero power equals 1. Question 16. Which of the following expressions is a perfect square trinomial? A) (x^2 + 6x + 9) B) (x^2 - 5x + 6) C) (x^2 + 2x + 1) D) Both A and C Answer: D Explanation: Both factor to ((x+3)^2) and ((x+1)^2) respectively. Question 17. If (f(x)=3x^2-2x+5), what is (f(-1))? A) 0 B) 10 C) 6 D) 4 Answer: C Explanation: Substitute (- 1 ): (3(1)-2(-1)+5 = 3+2+5 =10). Wait that gives 10, not 6. Let's recalc: Actually (3(-1)^2 = 3). (-2(-1)=2). So (3+2+5=10). The correct answer is B. Answer: B Explanation: Substituting (- 1 ) yields (3(1)+2+5=10). Question 18. Which exponent rule is used to rewrite ( \sqrt[3]{a^6}) as a power of (a)? A) Negative exponent rule B) Rational exponent rule C) Zero exponent rule D) Power‑of‑a‑product Answer: B Explanation: (\sqrt[3]{a^6}=a^{6/3}=a^{2}).
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Question 19. What is the result of adding (4x^3 + 2x^2) and (-3x^3 + 5x^2)? A) (x^3 + 7x^2) B) (7x^5) C) (x^3 - 3x^2) D) (x^5 + 7x^2) Answer: A Explanation: Combine like terms: (4x^3-3x^3 = x^3); (2x^2+5x^2 = 7x^2). Question 20. Which of the following is the correct FOIL expansion of ((2x+3)(x-4))? A) (2x^2 - 5x - 12 ) B) (2x^2 - 5x +12) C) (2x^2 - 8x +3x - 12 ) D) (2x^2 - 8x +3x - 12 ) (same as C) Answer: D Explanation: FOIL gives (2x·x = 2x^2), (2x·(-4) = - 8x), (3·x = 3x), (3·(-4) = - 12 ); combine to (2x^2 - 5x - 12 ). Actually after combining, the final simplified form is (2x^2 - 5x - 12 ). The answer choice that matches the unsimplified FOIL is C/D. Since the question asks for the correct FOIL expansion (unsimplified), both C and D are identical; choose D. Question 21. What is the degree of the polynomial (7)? A) 0 B) 1 C) Undefined D) 7 Answer: A Explanation: A constant has degree 0. Question 22. Simplify ((x^{-2} y^3)^2). A) (x^{-4} y^6) B) (x^{4} y^{6}) C) (\frac{y^6}{x^4}) D) Both A and C Answer: D Explanation: Square each factor: ((x^{-2})^2 = x^{-4}); ((y^3)^2 = y^6); writing with positive exponents gives (\frac{y^6}{x^4}). Question 23. Which of the following statements is true about the zero exponent rule?
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Question 28. What is the coefficient of (x^2) in the product ((x+2)(3x^2 - x + 5))? A) 3 B) 5 C) 6 D) 7 Answer: A Explanation: Multiply and collect (x^2) terms: (x·(-x) = - x^2) and (2·3x^2 = 6x^2); sum = (5x^2). Wait that gives 5. Let's recompute: Terms that give (x^2): (x·(-x) = - x^2); (2·3x^2 = 6x^2). Total (5x^2). So coefficient is 5. Answer: B Explanation: After expansion, the (x^2) term’s coefficient is 5. Question 29. Which of the following is equivalent to ( (x^5 y^{-2})^{-3})? A) (x^{-15} y^{6}) B) (x^{15} y^{-6}) C) (\frac{y^{6}}{x^{15}}) D) Both A and C Answer: D Explanation: Apply negative exponent: ((a^m)^n = a^{mn}); exponents become (- 15 ) for x and (6) for y; rewrite with positive exponents as (\frac{y^{6}}{x^{15}}). Question 30. What is the remainder when (x^3 + 4x^2 - x + 6) is divided by (x - 2 ) using synthetic division? A) 0 B) 2 C) 10 D) 14 Answer: C Explanation: Synthetic division with root 2 gives remainder 10. Question 31. If (a^2 b^3 = 72) and (a=2), find (b). A) 3 B) 4 C) 6 D) 9 Answer: C
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Explanation: Substitute (a=2): (4b^3 = 72) ⇒ (b^3 = 18) ⇒ (b = \sqrt[3]{18} ≈ 2.62). None of the options match; perhaps mis‑calculation. Let's check: Actually 4b^3=72 ⇒ b^3=18 ⇒ b≈2.62, not integer. Maybe the intended answer is 3 (since 3^3=27; 4*27=108). Something off. Let's replace question: Instead let (a=3). Then (9b^3=72) ⇒ (b^3=8) ⇒ (b=2). Not in options. To keep consistency, we will change numbers: Suppose (a=2) and product equals 16: (4b^3=16) ⇒ (b^3=4) ⇒ (b≈1.587). Not good. Let's discard this question and replace with a clearer one. Revised Question 31. If ( (2x)^3 = 64), what is the value of (x)? A) 2 B) 4 C) 8 D) 16 Answer: B Explanation: ((2x)^3 = 8x^3 = 64) ⇒ (x^3 = 8) ⇒ (x = 2). Wait that gives 2. Actually (8x^3 =64) ⇒ (x^3=8) ⇒ (x=2). So answer is A. Answer: A Explanation: Solving gives (x=2). Question 32. Which of the following is NOT a valid use of the product rule? A) (a^3·a^5 = a^8) B) (2^4·2^2 = 2^6) C) ((xy)^2 = x^2 y^2) D) (b^0·b^3 = b^3) Answer: C Explanation: The product rule applies to like bases, not to a product inside parentheses; ((xy)^2) uses power‑of‑a‑product rule. Question 33. What is the leading term of the product ((3x^2 - 5)(2x^3 + x))? A) (6x^5) B) (6x^4) C) ( - 10x^2) D) ( - 15x^5) Answer: A Explanation: Multiply highest‑degree terms: (3x^2·2x^3 = 6x^5).
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A) (x^3 + 6x^2 + 12x + 8) B) (x^3 + 8) C) (x^3 + 3x^2 + 3x + 2) D) (x^3 + 6x + 8) Answer: A Explanation: Use binomial theorem: coefficients 1,3,3,1 with powers of 2. Question 38. If (a^{1/2}=5), what is the value of (a)? A) 10 B) 25 C) 5 D) (\frac{1}{5}) Answer: B Explanation: Square both sides: (a = 5^2 = 25). Question 39. Which rule is applied when simplifying (\frac{x^7 y^3}{x^2 y^5})? A) Product rule B) Quotient rule C) Power‑of‑a‑product D) Negative exponent rule Answer: B Explanation: Subtract exponents for each base: (x^{7-2}=x^5), (y^{3-5}=y^{-2}) → (\frac{x^5}{y^2}). Question 40. Find the sum of the coefficients of (3x^4 - 2x^3 + 7x - 5 ). A) 3 B) 0 C) 5 D) - 5 Answer: B Explanation: Substitute (x=1): (3-2+7-5 = 3). Wait that equals 3. Let's compute: 3 - 2 =1; 1+7=8; 8-5=3. So answer is 3, which is option A. Answer: A Explanation: Evaluating at (x=1) gives the sum of coefficients, which is 3. Question 41. Which of the following is the correct factorization of (x^2 - 9 )? A) ((x-3)(x+3)) B) ((x-9)(x+1)) C) ((x-9)(x+9)) D) ((x-3)^2)
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Answer: A Explanation: Difference of squares. Question 42. If (p(x)=2x^3 - 6x^2 + 4x), factor completely. A) (2x(x-1)(x-2)) B) (2x(x^2 - 3x + 2)) C) (2(x-1)(x-2)) D) (2x^2(x-3)) Answer: B Explanation: Factor out GCF (2x); remaining quadratic factors to ((x-1)(x-2)) but the product is (x^2- 3x+2). So option B is the partially factored form; complete factorization is (2x(x-1)(x-2)) which matches A. Actually A is fully factored. Answer: A Explanation: After extracting (2x), the quadratic splits into ((x-1)(x-2)). Question 43. Which expression represents ((\frac{2}{3})^{ - 4})? A) (\frac{16}{81}) B) (\frac{81}{16}) C) (-\frac{16}{81}) D) (-\frac{81}{16}) Answer: B Explanation: Negative exponent inverts the fraction; ((3/2)^4 = 81/16). Question 44. What is the constant term in the expansion of ((x+5)^4)? A) 5 B) 25 C) 125 D) 625 Answer: D Explanation: Constant term is (5^4 = 625). Question 45. If (a^3 = 27) and (a>0), what is (a)? A) 3 B) 9 C) 6 D) 27
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Answer: D Explanation: Substituting (x = - 1 ) yields remainder 6. Question 50. If ((x^2)^3 = x^k), what is (k)? A) 5 B) 6 C) 8 D) 9 Answer: D Explanation: Multiply exponents: (2·3 = 6). Actually that gives 6, not 9. Wait: ((x^2)^3 = x^{2·3}=x^6). So answer is B. Answer: B Explanation: The exponent becomes 6. Question 51. Which of the following is NOT a special product? A) Difference of squares B) Perfect square trinomial C) Sum of cubes D) Linear binomial product Answer: D Explanation: “Linear binomial product” is not a standard named special product. Question 52. Simplify ( (4x^{-1} y^2)^3). A) (64 x^{-3} y^6) B) (64 x^{3} y^{6}) C) (\frac{64 y^6}{x^3}) D) Both A and C Answer: D Explanation: Cube each factor; write with positive exponents as (\frac{64 y^6}{x^3}). Question 53. Find the coefficient of (x^3) in the expansion of ((2x - 1)^4). A) (- 8 ) B) (- 16 ) C) (8) D) (16) Answer: B
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Explanation: Binomial coefficients: (C(4,1)=4); term is (4(2x)^3(-1) = 4·8x^3·(-1) = - 32x^3). Wait coefficient is - 32 not listed. Let's recompute: Actually term with (x^3) comes from choosing 3 copies of (2x) and 1 copy of (- 1 ). Coefficient = (C(4,1)(2)^3(-1) = 48(-1) = - 32 ). Not in options. Replace with a different exponent. Revised Question 53. Find the coefficient of (x^2) in the expansion of ((3x + 2)^3). A) 27 B) 54 C) 108 D) 162 Answer: B Explanation: Term with (x^2): choose 2 copies of (3x) and 1 copy of 2: coefficient = (C(3,1)(3)^2 = 392 = 54). Question 54. Which expression is equivalent to (\frac{1}{a^{-4}})? A) (a^4) B) (a^{-4}) C) (\frac{1}{a^4}) D) (-a^4) Answer: A Explanation: Reciprocal of a negative exponent makes the exponent positive. Question 55. If (f(x)=x^4 - 5x^2 + 4), which of the following is a factor? A) ((x-1)) B) ((x+2)) C) ((x^2-1)) D) ((x^2+4)) Answer: C Explanation: Substitute (x=1) gives 0? (1-5+4=0) yes, so ((x-1)) factor, but also ((x+1)) factor; product ((x^2-1)) is a factor. Question 56. Simplify ((\frac{x^3}{y^2})^{-2}). A) (\frac{x^6}{y^4}) B) (\frac{y^4}{x^6}) C) (\frac{x^{-6}}{y^{-4}}) D) (\frac{y^{-4}}{x^{-6}}) Answer: B
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A) (a^{3/4}) B) (a^{4/3}) C) (a^{3}) D) (a^{1/4}) Answer: A Explanation: (\sqrt[4]{a^3}=a^{3/4}). Question 62. If (h(x)=x^2 + 6x + 9), what is (h(-3))? A) 0 B) 9 C) 36 D) - 9 Answer: A Explanation: Plugging (- 3 ): (9 - 18 +9 =0). Question 63. Which of the following correctly uses the quotient rule? A) (\frac{a^5}{a^2}=a^{7}) B) (\frac{b^4}{b}=b^{3}) C) (\frac{c^0}{c^3}=c^{-3}) D) Both B and C Answer: D Explanation: B and C follow exponent subtraction. Question 64. What is the coefficient of (x) in the product ((2x-5)(3x+4))? A) 8 B) - 22 C) 22 D) - 8 Answer: B Explanation: Expand: (2x·4 = 8x); (-5·3x = - 15x); sum = (-7x). Wait coefficient is - 7 not listed. Let's recompute full expansion: (2x·3x = 6x^2); (2x·4=8x); (-5·3x = - 15x); (-5·4 = - 20 ). Combine x terms: (8x-15x = - 7x). Not in options. Replace with different numbers. Revised Question 64. What is the coefficient of (x) in the product ((x+3)(2x-5))? A) - 1 B) 1 C) - 2 D) 2 Answer: A
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Explanation: Expand: (x·(-5) = - 5x); (3·2x = 6x); sum = (x). Wait that gives 1, not - 1. Let's compute: ((x+3)(2x-5) = 2x^2 - 5x +6x - 15 = 2x^2 + x - 15 ). Coefficient of (x) is 1. So answer B. Answer: B Explanation: After expansion the linear term is (+1x). Question 65. Which of the following is the correct factorization of (x^2 + 5x + 6)? A) ((x+2)(x+3)) B) ((x-2)(x-3)) C) ((x+1)(x+6)) D) ((x-1)(x-6)) Answer: A Explanation: Product of 2 and 3 gives 6 and sum 5. Question 66. If ((a^3 b^{-2})^2 = a^k b^m), what are (k) and (m)? A) (k=6, m=- 4 ) B) (k=6, m=4) C) (k=-6, m=4) D) (k=-6, m=- 4 ) Answer: A Explanation: Multiply exponents: (3·2=6); (-2·2=- 4 ). Question 67. What is the remainder when (x^4 - 2x^2 + 1) is divided by (x^2 + 1)? A) 0 B) (-3x^2) C) (2x^2) D) (-x^2 + 1) Answer: A Explanation: Perform division: (x^4 - 2x^2 +1 = (x^2+1)(x^2 - 1) +0). Question 68. Which expression correctly applies the power‑of‑a‑product rule to ((3xy)^2)? A) (9x^2 y^2) B) (6xy) C) (3x^2 y) D) (9xy) Answer: A
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Explanation: Highest degree term with its coefficient. Question 73. If (a^{1/3}=2), find (a). A) 4 B) 6 C) 8 D) 16 Answer: C Explanation: Cube both sides: (a = 2^3 = 8). Question 74. Which of the following is the result of applying the product rule to ( (5x^2)^3)? A) (125x^5) B) (125x^6) C) (15x^5) D) (15x^6) Answer: B Explanation: Raise coefficient and variable: (5^3 =125); (x^{2·3}=x^6). Question 75. Find the sum of the coefficients of ((2x - 3)^3). A) (- 1 ) B) 1 C) 7 D) - 7 Answer: D Explanation: Evaluate at (x=1): ((2·1-3)^3 = (-1)^3 = - 1 ). Wait sum is - 1, which is option A. Actually option A is - 1. Answer: A Explanation: Substituting (x=1) gives ((-1)^3 = - 1 ). Question 76. Which expression is equivalent to (\frac{1}{(x^{-2} y^3)})? A) (x^{2} y^{-3}) B) (x^{-2} y^{3}) C) (\frac{x^{2}}{y^{3}}) D) (\frac{y^{3}}{x^{2}}) Answer: C Explanation: Reciprocal flips signs of exponents: becomes (x^{2} / y^{3}).
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Question 77. If (p(x)=x^3 - 4x), factor completely. A) (x(x^2 - 4)) B) (x(x-2)(x+2)) C) Both A and B D) None of the above Answer: C Explanation: First factor out (x), then difference of squares. Question 78. Which of the following is a perfect square trinomial? A) (x^2 + 10x + 25) B) (x^2 - 6x + 9) C) (x^2 + 2x + 1) D) All of the above Answer: D Explanation: All factor to ((x+5)^2, (x-3)^2, (x+1)^2). Question 79. Simplify ((\frac{2x^3}{y^2})^{-1}). A) (\frac{y^2}{2x^3}) B) (\frac{2x^3}{y^2}) C) (\frac{x^3}{2y^2}) D) (\frac{y^2}{x^3}) Answer: A Explanation: Negative exponent inverts fraction. Question 80. What is the degree of the term (-7x^{0} y^{5})? A) 0 B) 5 C) 7 D) Not a polynomial term Answer: B Explanation: Degree is sum of exponents of variables; (0+5=5). Question 81. If (f(x)=3x^2 - 12x + 12), what is the value of (f(2))? A) 0 B) 4 C) 6 D) 12
