Grade 9 Algebra — Expressions Practice — Medium

Question 1 of 40Score 0/0Medium

Question 1 of 40: Which is NOT a polynomial?

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Keep going: Read the lesson: Polynomial Expressions Scientific Notation Exponent Rules
Answer key for parents & teachers (40 questions)
  1. Which is NOT a polynomial? — x⁻² + 4. x⁻² has a negative exponent — not allowed in polynomials. All others are valid polynomials.
  2. What is the sum (2x² + 3x − 1) + (x² − x + 4)? — 3x² + 2x + 3. Combine like terms: (2+1)x² + (3−1)x + (−1+4) = 3x² + 2x + 3.
  3. What is (4x² − 3x) − (x² + 5x − 2)? — 3x² − 8x + 2. Distribute the minus: 4x²−3x − x²−5x+2 = (4−1)x² + (−3−5)x + 2 = 3x² − 8x + 2.
  4. Factor out the GCF from 15x⁴ − 10x³ + 5x². — 5x²(3x² − 2x + 1). GCF = 5x². Dividing: 15x⁴/5x²=3x², 10x³/5x²=2x, 5x²/5x²=1. Result: 5x²(3x²−2x+1).
  5. A monomial is a polynomial with how many terms? — Exactly 1. Monomial = 1 term; binomial = 2 terms; trinomial = 3 terms. Polynomial is the general name for all.
  6. What is the degree of the constant polynomial 5? — 0. A non-zero constant = 5 = 5x⁰. The degree is 0. The polynomial 0 has undefined (or −∞) degree by convention.
  7. Simplify: 3x(x² − 2x + 4). — 3x³ − 6x² + 12x. Distribute: 3x·x²=3x³, 3x·(−2x)=−6x², 3x·4=12x. Result: 3x³ − 6x² + 12x.
  8. Evaluate P(0) for P(x) = 5x³ − 3x² + x − 7. — −7. P(0) = 5(0) − 3(0) + 0 − 7 = −7. P(0) always equals the constant term.
  9. Factor x² + 5x − 6. — (x−1)(x+6). Need numbers that multiply to −6 and add to 5: −1 and 6. (x−1)(x+6). Check: −1+6=5 ✓, −1·6=−6 ✓.
  10. Factor 2x² + 7x + 3 using the AC method. — (2x+1)(x+3). AC = 2·3 = 6. Factors of 6 that sum to 7: 1 and 6. Rewrite: 2x²+x+6x+3 = x(2x+1)+3(2x+1) = (x+3)(2x+1).
  11. Which pattern represents a perfect square trinomial? — a² + 2ab + b² = (a+b)². Perfect square trinomial: a² + 2ab + b² = (a+b)² or a² − 2ab + b² = (a−b)².
  12. Factor 4x² − 25. — (2x−5)(2x+5). Difference of squares: 4x²−25 = (2x)²−5² = (2x−5)(2x+5).
  13. Factor 3x² − 12x. — 3x(x−4). Factor out GCF = 3x: 3x(x−4).
  14. Find the roots of x² + x − 6 = 0 by factoring. — x = 2 and x = −3. x²+x−6 = (x+3)(x−2) = 0. Roots: x = 2 and x = −3. Check: (2+3)(2−2)=0 ✓.
  15. Factor x² − x − 20. — (x−5)(x+4). Need numbers multiplying to −20 adding to −1: −5 and 4. (x−5)(x+4). Check: −5+4=−1 ✓, −5·4=−20 ✓.
  16. What is the zero of the factor (2x − 6)? — x = 3. 2x − 6 = 0 → 2x = 6 → x = 3.
  17. Why do we "verify by expanding" after factoring? — To ensure the factored form multiplies back to the original. One sign error in factoring changes the entire result. Expanding (FOIL) back to the original is the only reliable check.
  18. Simplify (2x²)³. — 8x⁶. (2x²)³ = 2³ · (x²)³ = 8 · x⁶ = 8x⁶.
  19. Simplify x⁻² · x⁵. — x³. Product rule: x⁻² · x⁵ = x^(−2+5) = x³.
  20. Simplify (x⁴y²)³. — x¹²y⁶. (x⁴y²)³ = x^(4·3) · y^(2·3) = x¹²y⁶.
  21. What is (x/y)³? — x³/y³. (x/y)³ = x³/y³. The power distributes to both numerator and denominator.
  22. Simplify x⁵/x⁸. — x⁻³ = 1/x³. Quotient rule: x⁵/x⁸ = x^(5−8) = x⁻³ = 1/x³.
  23. Simplify (3x²y⁻¹)². — 9x⁴/y². (3x²y⁻¹)² = 3² · x⁴ · y⁻² = 9x⁴/y².
  24. What is (ab)⁵? — a⁵b⁵. A product raised to a power: (ab)⁵ = a⁵ · b⁵. The power applies to each factor.
  25. Simplify: x³ · x⁻⁷. — x⁻⁴. x³ · x⁻⁷ = x^(3+(−7)) = x⁻⁴ = 1/x⁴.
  26. Express 3^(−2) as a fraction. — 1/9. 3^(−2) = 1/3² = 1/9.
  27. Simplify (xy)⁰. — 1. Any nonzero expression raised to the power 0 equals 1. (xy)⁰ = 1 (assuming x≠0 and y≠0).
  28. Which of the following equals x^(1/2) · x^(1/2)? — x. x^(1/2) · x^(1/2) = x^(1/2+1/2) = x¹ = x.
  29. Add: (2x² − 3x + 1) + (x² + 5x − 4). — 3x² + 2x − 3. (2+1)x² + (−3+5)x + (1−4) = 3x² + 2x − 3.
  30. Subtract: (3x² + 4x − 5) − (x² − 2x + 3). — 2x² + 6x − 8. Distribute −: 3x²+4x−5−x²+2x−3 = 2x²+6x−8.
  31. What is (x − 3)²? — x² − 6x + 9. (x−3)² = (x−3)(x−3). FOIL: x²−3x−3x+9 = x²−6x+9.
  32. Multiply: 2x(3x² − 5x + 4). — 6x³ − 10x² + 8x. 2x·3x²=6x³; 2x·(−5x)=−10x²; 2x·4=8x. Result: 6x³−10x²+8x.
  33. What is (2x + 3)(x − 1)? — 2x² + x − 3. F: 2x², O: −2x, I: 3x, L: −3. Sum: 2x² + x − 3.
  34. When subtracting (2x − 3) from (5x + 1), what do you get? — 3x + 4. (5x+1) − (2x−3) = 5x+1−2x+3 = 3x+4.
  35. Expand (x + a)²: — x² + 2ax + a². (x+a)² = x² + 2ax + a². This is the perfect square trinomial formula.
  36. What does the "Like Terms" rule require? — Same variable AND same exponent. Like terms must have exactly the same variable(s) raised to exactly the same power(s). Only their coefficients can differ.
  37. Can x² and x³ be combined by adding? — No, they are not like terms. x² and x³ are NOT like terms — the exponents differ. They cannot be combined by addition/subtraction.
  38. What is the constant term in the expansion of (x + 3)(x − 2)? — −6. L in FOIL: 3 · (−2) = −6. The constant term is −6.
  39. What is wrong with: (x + 2)² = x² + 4? — The middle term 2·x·2 = 4x is missing. (x+2)² = x² + 2(x)(2) + 4 = x² + 4x + 4. The middle term 4x is missing in the wrong version.
  40. Simplify: (3x + 2y) − 2(x − y). — x + 4y. 3x + 2y − 2x + 2y = (3−2)x + (2+2)y = x + 4y.