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Term pack

Grade 9 Math

Name: ________________________

Skills in this pack

  • Algebraic identities
  • Factorisation
  • Polynomials
  • Quadratic equations
  • Arithmetic progressions

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Grade 9 Math · 1 of 5

Algebraic identities

Use (a + b)², (a - b)² and a² - b² to expand brackets and to multiply quickly.

  1. 1.Expand (a + b)².

    • a) a² + 2ab + b²
    • b) 2a + 2b
    • c) a² + b²
    • d) a² - 2ab + b²

  2. 2.Match each product to its expansion.

    • (x + 1)²
    • (x - 1)²
    • (x + 2)(x - 2)
    • (x + 4)²
    • x² - 2x + 1
    • x² + 2x + 1
    • x² - 4
    • x² + 8x + 16

  3. 3.Expand (x + 3)².

    • a) x² + 9
    • b) x² + 6x + 6
    • c) x² + 6x + 9
    • d) x² + 3x + 9

  4. 4.Use (100 + 2)² to work out 102².

  5. 5.Use a² - b² = (a + b)(a - b) to work out 43² - 37².

  6. 6.Use (100 - 2)² to work out 98².

  7. 7.Use a² - b² = (a + b)(a - b) to work out 105 × 95.

  8. 8.Expand (a - b)².

    • a) a² + 2ab + b²
    • b) a² - b²
    • c) a² + b²
    • d) a² - 2ab + b²

Algebraic identities · Grade 9 Math · www.arenapublications.com/learn

Grade 9 Math · 2 of 5

Factorisation

Break an algebraic expression into factors, and use those factors to divide one expression by another.

  1. 1.What is the highest common factor of the two terms in 12x + 18?

  2. 2.Sort each expression by the method that factorizes it.

    Groups: Take out a common factor · Use a² - b² = (a - b)(a + b)

    • 7x + 21
    • 9a² - 4b²
    • 12m + 18n
    • x² - 1
    • 5y² - 15y
    • x² - 25

  3. 3.x² + 7x + 12 factorizes as (x + 3)(x + ?). What number belongs in place of the question mark?

  4. 4.Take a out of a² + ab as a common factor. What is left inside the bracket?

  5. 5.Which of these is x² + 10x + 25 written in factorized form?

    • a) (x + 5)²
    • b) (x + 10)²
    • c) (x + 5)(x - 5)
    • d) (x + 25)(x + 1)

  6. 6.Which of these is 4x² - 9 written in factorized form?

    • a) (2x - 3)(2x - 3)
    • b) (4x - 3)(x + 3)
    • c) (2x - 9)(2x + 1)
    • d) (2x - 3)(2x + 3)

  7. 7.Put these expressions in order by the highest common factor of their terms, smallest first.

    • 14x + 21
    • 6x + 18
    • 10x + 25
    • 4x + 8
    • 3x + 9

  8. 8.Match each expression to its factorized form.

    • x² - 16
    • x² + 8x + 16
    • 3x + 12
    • x² - 4x
    • 2x² - 32
    • (x + 4)²
    • (x - 4)(x + 4)
    • x(x - 4)
    • 3(x + 4)
    • 2(x - 4)(x + 4)

Factorisation · Grade 9 Math · www.arenapublications.com/learn

Grade 9 Math · 3 of 5

Polynomials

Find the degree of a polynomial, evaluate it at a value and recognize its zeroes.

  1. 1.If p(x) = 2x + 6, what value of x makes p(x) = 0?

  2. 2.If p(x) = x² - 3x + 2, what is p(0)?

  3. 3.If p(x) = x³ - 2x² + x, what is p(1)?

  4. 4.Match each polynomial to its degree.

    • 7
    • 4x + 1
    • x² - 3
    • 2x³ + x
    • x⁵
    • 1
    • 5
    • 2
    • 3
    • 0

  5. 5.What is the degree of the polynomial 5x + 1?

  6. 6.The factor theorem says that if p(k) = 0 then (x - k) is a factor of p(x). Given that p(2) = 0, what is k?

  7. 7.If p(x) = x² - 3x + 2, what is p(2)?

  8. 8.A polynomial of degree 2 is called a...

    • a) linear polynomial
    • b) quadratic polynomial
    • c) constant polynomial
    • d) cubic polynomial

Polynomials · Grade 9 Math · www.arenapublications.com/learn

Grade 9 Math · 4 of 5

Quadratic equations

Solve quadratic equations by factorizing, and use the discriminant to describe the roots.

  1. 1.A quadratic equation whose discriminant is 0 has...

    • a) three real roots
    • b) two different real roots
    • c) two equal real roots
    • d) no real roots

  2. 2.Solve x² + 7x + 12 = 0. Give the smaller root.

  3. 3.Solve x² - 5x + 6 = 0. Give the smaller root.

  4. 4.What is the discriminant, b² - 4ac, of x² - 3x + 5 = 0?

  5. 5.Solve x² - 9 = 0. Give the positive root.

  6. 6.Solve x² - 4x - 5 = 0. Give the positive root.

  7. 7.For x² - 7x + 10 = 0, what is the product of the two roots?

  8. 8.Which of these is a quadratic equation?

    • a) x + 7 = 0
    • b) 2x + 5 = 0
    • c) x² - 4x + 3 = 0
    • d) x³ - 1 = 0

Quadratic equations · Grade 9 Math · www.arenapublications.com/learn

Grade 9 Math · 5 of 5

Arithmetic progressions

Find the common difference, the nth term and the sum of an arithmetic progression.

Before you start

This line is the progression 5, 10, 15, 20 and onwards. Move from one term to the next and count the steps you have taken.

510152025303540

The first term takes no steps at all, so the 8th term is 5 with seven steps of 5 added on: 5 + 7 × 5 = 40. That is a + (n - 1)d, read straight off the line.

  1. 1.Find the sum 1 + 2 + 3 + ... + 100.

  2. 2.The first term of an AP is 7 and the common difference is 5. What is the 4th term?

  3. 3.Which of these lists is an arithmetic progression?

    • a) 2, 5, 8, 11
    • b) 1, 2, 4, 8
    • c) 1, 4, 9, 16
    • d) 3, 3, 4, 6

  4. 4.In the AP 5, 8, 11, ... what is the 20th term?

  5. 5.In the AP 3, 7, 11, 15, ... what is the 10th term?

  6. 6.Find the sum of the first 15 terms of 1, 3, 5, 7, ...

  7. 7.Match each progression to its common difference.

    • 2, 5, 8, 11
    • 1, 3, 5, 7
    • 10, 6, 2, -2
    • 7, 8, 9, 10
    • 0, 10, 20, 30
    • 1
    • 2
    • 10
    • -4
    • 3

  8. 8.In the AP 20, 17, 14, ... what is the common difference?

Arithmetic progressions · Grade 9 Math · www.arenapublications.com/learn

Answer keys — Grade 9 Math

In the same order as the worksheets.

Algebraic identities

  1. 1. a) a² + 2ab + b²
  2. 2. (x + 1)² → x² + 2x + 1; (x - 1)² → x² - 2x + 1; (x + 2)(x - 2) → x² - 4; (x + 4)² → x² + 8x + 16
  3. 3. c) x² + 6x + 9
  4. 4. 10404
  5. 5. 480
  6. 6. 9604
  7. 7. 9975
  8. 8. d) a² - 2ab + b²

Factorisation

  1. 1. 6
  2. 2. 7x + 21 → Take out a common factor; x² - 25 → Use a² - b² = (a - b)(a + b); 5y² - 15y → Take out a common factor; 9a² - 4b² → Use a² - b² = (a - b)(a + b); 12m + 18n → Take out a common factor; x² - 1 → Use a² - b² = (a - b)(a + b)
  3. 3. 4
  4. 4. a + b
  5. 5. a) (x + 5)²
  6. 6. d) (2x - 3)(2x + 3)
  7. 7. 1. 3x + 9 2. 4x + 8 3. 10x + 25 4. 6x + 18 5. 14x + 21
  8. 8. x² - 16 → (x - 4)(x + 4); x² + 8x + 16 → (x + 4)²; 3x + 12 → 3(x + 4); x² - 4x → x(x - 4); 2x² - 32 → 2(x - 4)(x + 4)

Polynomials

  1. 1. -3
  2. 2. 2
  3. 3. 0
  4. 4. 7 → 0; 4x + 1 → 1; x² - 3 → 2; 2x³ + x → 3; x⁵ → 5
  5. 5. 1
  6. 6. 2
  7. 7. 0
  8. 8. b) quadratic polynomial

Quadratic equations

  1. 1. c) two equal real roots
  2. 2. -4
  3. 3. 2
  4. 4. -11
  5. 5. 3
  6. 6. 5
  7. 7. 10
  8. 8. c) x² - 4x + 3 = 0

Arithmetic progressions

  1. 1. 5050
  2. 2. 22
  3. 3. a) 2, 5, 8, 11
  4. 4. 62
  5. 5. 39
  6. 6. 225
  7. 7. 2, 5, 8, 11 → 3; 1, 3, 5, 7 → 2; 10, 6, 2, -2 → -4; 7, 8, 9, 10 → 1; 0, 10, 20, 30 → 10
  8. 8. -3

Grade 9 Math · www.arenapublications.com/learn