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About Algebraic Expressions — Class 8 Olympiad

Add, subtract, multiply algebraic expressions; apply identities and solve expression-based competition problems. This topic is part of the Olympiad Class 8 mathematics syllabus (chapter: Module 5). On this page you can practice 57 questions across three difficulty levels — 19 easy, 20 medium, and 18 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Algebraic Expressions — solved examples for Class 8 Olympiad

Example 1easy

If x + 1/x = 4, what is the value of x⁴ + 1/x⁴?
  1. A)194
  2. B)196
  3. C)256
  4. D)260

Step-by-step solution

  1. Given x + 1/x = 4. Squaring both sides:
  2. (x + 1/x)² = 4² => x² + 2(x)(1/x) + (1/x)² = 16 => x² + 2 + 1/x² = 16 => x² + 1/x² = 14.
  3. Now, square (x² + 1/x²) = 14:
  4. (x² + 1/x²)² = 14² => (x²)² + 2(x²)(1/x²) + (1/x²)² = 196 => x⁴ + 2 + 1/x⁴ = 196 => x⁴ + 1/x⁴ = 194.

Answer: 194

Example 2medium

If x² - 5x + 1 = 0, then what is the value of x⁴ + 1/x⁴?
  1. A)527
  2. B)525
  3. C)529
  4. D)523

Step-by-step solution

  1. Given x² - 5x + 1 = 0. Since x cannot be 0, divide by x: x - 5 + 1/x = 0, which implies x + 1/x = 5.
  2. Square both sides: (x + 1/x)² = 5² => x² + 2(x)(1/x) + (1/x)² = 25 => x² + 2 + 1/x² = 25 => x² + 1/x² = 23.
  3. Square both sides again: (x² + 1/x²)² = 23² => (x²)² + 2(x²)(1/x²) + (1/x²)² = 529 => x⁴ + 2 + 1/x⁴ = 529.
  4. Therefore, x⁴ + 1/x⁴ = 529 - 2 = 527.

Answer: 527

Example 3hard

If x + 1/x = 5, what is the value of x⁴ + 1/x⁴?
  1. A)527
  2. B)625
  3. C)525
  4. D)500

Step-by-step solution

  1. Given x + 1/x = 5. Squaring both sides: (x + 1/x)² = 5²
    x2+2(x)(1/x)+1/x2=25x² + 2(x)(1/x) + 1/x² = 25
  2. This simplifies to x² + 2 + 1/x² = 25, so x² + 1/x² = 23.
  3. Now, square (x² + 1/x² = 23) again: (x² + 1/x²)² = 23²
    (x2)2+2(x2)(1/x2)+(1/x2)2=529(x²)² + 2(x²)(1/x²) + (1/x²)² = 529
  4. This simplifies to x⁴ + 2 + 1/x⁴ = 529. Therefore, x⁴ + 1/x⁴ = 529 - 2 = 527.

Answer: 527

Practice questions on Algebraic Expressions

  1. Q1.easy

    In the expansion of (3x² - 5x + 1)(2x² + 4x - 3), what is the coefficient of x³?
    1. A)-17
    2. B)-13
    3. C)13
    4. D)17
    Show answer

    Answer: -17

    Hint: Identify all pairs of terms from the two polynomials whose product results in an x³ term. Then sum their coefficients.

  2. Q2.easy

    What is the exact value of (99.5)² - (0.5)²?
    1. A)9900
    2. B)9950
    3. C)99.5
    4. D)100
    Show answer

    Answer: 9950

    Hint: This problem uses a fundamental algebraic identity, a² - b² = (a - b)(a + b).

  3. Q3.easy

    Simplify the expression (p + q - r)² - (p - q + r)².
    1. A)4pq - 4pr
    2. B)4pr - 4pq
    3. C)4pq + 4pr
    4. D)4qr - 4pr
    Show answer

    Answer: 4pq - 4pr

    Hint: Treat this as a² - b² = (a - b)(a + b) where a = (p + q - r) and b = (p - q + r).

  4. Q4.medium

    What is the value of (x - 1)(x + 1)(x² + 1)(x⁴ + 1)(x⁸ + 1) when x = 2?
    1. A)2¹⁵ - 1
    2. B)2¹⁶ - 1
    3. C)2¹⁶ + 1
    4. D)2¹⁵ + 1
    Show answer

    Answer: 2¹⁶ - 1

    Hint: Repeatedly apply the difference of squares identity (a - b)(a + b) = a² - b². Start from the first two terms.

  5. Q5.medium

    If (2x + A)(Bx - 3) = Cx² - x - 6 for all values of x, find the value of A + B + C.
    1. A)4
    2. B)5
    3. C)6
    4. D)7
    Show answer

    Answer: 6

    Hint: Expand the left side of the equation and then compare the coefficients of corresponding powers of x with the right side.

  6. Q6.medium

    A rectangular garden has length (3x + 2y) units and breadth (3x - 2y) units. If its area is 117 square units and x = 5, what is the value of y?
    1. A)1
    2. B)2
    3. C)3
    4. D)4
    Show answer

    Answer: 3

    Hint: The area of a rectangle is length × breadth. Use the identity (a + b)(a - b) = a² - b² to simplify the area expression.

  7. Q7.hard

    Simplify the expression: (a² + b² - c² + 2ab)(c² - a² - b² + 2ab).
    1. A)(a+b)² - c⁴
    2. B)4a²b² - (c² - (a+b)²)²
    3. C)(a+b)² - c²
    4. D)4a²b² - (a² + b² - c²)²
    Show answer

    Answer: 4a²b² - (a² + b² - c²)²

    Hint: Rearrange terms to identify the difference of squares identity, X² - Y².

  8. Q8.hard

    If the expression (kx + 1/x)² = 9x² + M + 1/x² for all non-zero x, and k is a positive constant, what is the value of k + M?
    1. A)6
    2. B)7
    3. C)8
    4. D)9
    Show answer

    Answer: 9

    Hint: Expand the left side using the (a+b)² identity and compare the coefficients of corresponding terms with the right side.

  9. Q9.hard

    If x + y = 7 and xy = 10, find the value of x³ + y³.
    1. A)133
    2. B)243
    3. C)343
    4. D)70
    Show answer

    Answer: 133

    Hint: Recall the identity for the sum of cubes: a³ + b³ = (a+b)(a² - ab + b²).

These are 9 of the 57 questions available for Algebraic Expressions. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.