Algebraic manipulation Revision Notes

Simplifying an algebraic expression means rewriting it in its simplest form by combining like terms. Like terms have exactly the same variables raised to the same powers. Constants (numbers without variables) are also combined together.

For example, consider the expression: 2a + 3ab - 1 + 5a - 9ab + 4.

Step 1: Identify like terms: 2a and 5a are like terms because both contain 'a'. 3ab and -9ab are like terms because both contain 'ab'. -1 and 4 are constants and can be combined as well.

• Combine 2a and 5a: 2a + 5a = 7a
• Combine 3ab and -9ab: 3ab - 9ab = -6ab
• Combine -1 and 4: -1 + 4 = 3
• Final simplified expression: 7a - 6ab + 3

Expanding means multiplying out brackets to write the expression as a sum or difference of terms without brackets.

For a single bracket multiplied by a term, multiply the term by each term inside the bracket.

Example 1: Expand 3x(2x - 4y) by multiplying 3x by each term inside the bracket: 3x × 2x = 6x² and 3x × (-4y) = -12xy, so the expanded form is 6x² - 12xy.

• For two brackets, multiply each term in the first bracket by each term in the second bracket.
• Example 2: Expand (3x + y)(x - 4y).
• Multiply 3x by x: 3x × x = 3x²
• Multiply 3x by -4y: 3x × (-4y) = -12xy
• Multiply y by x: y × x = xy
• Multiply y by -4y: y × (-4y) = -4y²
• Combine like terms: 3x² - 12xy + xy - 4y² = 3x² - 11xy - 4y²
• For more than two brackets, expand two brackets first, then multiply the result by the next bracket.

Factoring is the reverse of expanding. It involves writing an expression as a product of factors. The simplest method is to take out the greatest common factor (GCF) from all terms.

Example: Factor 9x + 15xy by finding the GCF of 9x and 15xy.

Step 1: Identify the GCF of the coefficients 9 and 15, which is 3, and the common variable factor x (present in both terms). So, GCF is 3x.

• Divide each term by 3x: 9x ÷ 3x = 3, 15xy ÷ 3x = 5y
• Write the factored form: 3x(3 + 5y)
• For more complex expressions, factor by grouping or recognise special forms to factor completely.

Certain algebraic expressions follow patterns that allow specific factoring methods.

Examples include: ax + bx + kay + kby, ax - by, a + 2ab + b, ax + bx + c, and ax + bx + cx.

Recognising these forms helps to factor efficiently.

• For ax + bx + kay + kby, group terms: (ax + bx) + (kay + kby) = x(a + b) + ky(a + b) = (a + b)(x + ky)
• For ax - by, factor as a difference of terms, sometimes as (√a)² - (√b)² if applicable.
• For a + 2ab + b, recognise it as a perfect square trinomial: (a + b)²
• For ax + bx + c, factor by grouping or trial methods depending on the values.
• For ax + bx + cx, factor by taking out common factors stepwise.

Completing the square rewrites a quadratic expression ax² + bx + c into the form a(x + d)² + e, where d and e are constants. This form is useful for solving equations and graphing.

Step 1: If a ≠ 1, factor out a from the x² and x terms: ax² + bx = a(x² + (b/a)x)

Step 2: Find (b/2a)² and add and subtract it inside the bracket to complete the square: a[x² + (b/a)x + (b/2a)² - (b/2a)²] + c = a[(x + b/2a)² - (b/2a)²] + c = a(x + b/2a)² - a(b/2a)² + c = a(x + b/2a)² + (c - b²/4a)

• Example: Complete the square for 2x² + 8x + 5.
• Factor out 2: 2(x² + 4x) + 5
• Calculate (4/2)² = 2² = 4
• Add and subtract 4 inside the bracket: 2(x² + 4x + 4 - 4) + 5 = 2[(x + 2)² - 4] + 5
• Expand: 2(x + 2)² - 8 + 5 = 2(x + 2)² - 3
• Final form: 2(x + 2)² - 3