Expanding Brackets and Simplifying Expressions

Expanding Brackets and Simplifying Expressions

A Level Links

• Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices, and surds.

Key Points

• When expanding one set of brackets, multiply everything inside by what is outside.
• Expanding two linear expressions, each with two terms of the form ax + b (where a ≠ 0 and b ≠ 0) produces four terms. Two of them can typically be simplified by collecting like terms.

Examples

Example 1

• Expand:
  $4(3x - 2) = 12x - 8$
  Explanation: Multiply everything inside the bracket by the 4 outside the bracket.

Example 2

• Expand and Simplify:$3(x + 5) - 4(2x + 3)$
  • Step 1: Expand each set of brackets separately:
  • $3(x + 5) = 3x + 15$
  • $-4(2x + 3) = -8x - 12$
  • Step 2: Combine them:
    $3x + 15 - 8x - 12 = 3 - 5x$

Example 3

• Expand and Simplify:$(x + 3)(x + 2)$
  • Step 1: Multiply each term:
    $= x(x + 2) + 3(x + 2)$
    $= x^2 + 2x + 3x + 6$
  • Step 2: Combine like terms:
    $x^2 + 5x + 6$

Example 4

• Expand and Simplify:$(x - 5)(2x + 3)$
  • Step 1: Multiply out:
    $x(2x + 3) - 5(2x + 3) = 2x^2 + 3x - 10x - 15$
  • Step 2: Combine like terms:
    $2x^2 - 7x - 15$

Practice

1. Expand:

   • a) $3(2x - 1)$
   • b) $-2(5pq + 4q^2)$
   • c) $-(3xy - 2y^2)$

2. Expand and simplify:

   • a) $7(3x + 5) + 6(2x - 8)$
   • b) $8(5p - 2) - 3(4p + 9)$
   • c) $9(3s + 1) - 5(6s - 10)$
   • d) $2(4x - 3) - (3x + 5)$

3. Expand:

   • a) $3x(4x + 8)$
   • b) $4k(5k^2 - 12)$
   • c) $-2h(6h^2 + 11h - 5)$
   • d) $-3s(4s^2 - 7s + 2)$

4. Expand and simplify:

   • a) $3(y^2 - 8) - 4(y^2 - 5)$
   • b) $2x(x + 5) + 3x(x - 7)$
   • c) $4p(2p - 1) - 3p(5p - 2)$
   • d) $3b(4b - 3) - b(6b - 9)$

5. Expand:
   $(2y - 8) \ 1 \ 2$

6. Expand and simplify:

   • a) $13 - 2(m + 7)$
   • b) $5p(p^2 + 6p) - 9p(2p - 3)$

7. Write down an expression in terms of x for the area of the rectangle shown in the diagram.

• Show that the area can be expressed as $21x^2 - 35x$.

1. Expand and simplify:
   • a) $(x + 4)(x + 5)$
   • b) $(x + 7)(x + 3)$
   • c) $(x + 7)(x - 2)$
   • d) $(x + 5)(x - 5)$
   • e) $(2x + 3)(x - 1)$
   • f) $(3x - 2)(2x + 1)$
   • g) $(5x - 3)(2x - 5)$
   • h) $(3x - 2)(7 + 4x)$
   • i) $(3x + 4y)(5y + 6x)$
   • j) $(x + 5)^2$
   • k) $(2x - 7)^2$
   • l) $(4x - 3y)^2$

Additional Notes

• Watch out when multiplying or dividing by negative numbers; changing the sign is essential.

Answers

1. a) 6x - 3 b) -10pq - 8q^2 c) -3xy + 2y^2 …