Algebra · KS3

Simplify, Expand, Factorise

Three ways of rewriting an expression without changing its value. Simplifying tidies it. Expanding multiplies out a bracket. Factorising puts the bracket back. One picture, a rectangle, explains the last two, and shows why they are the same move run backwards.

National curriculum KS3: simplify and manipulate algebraic expressions by collecting like terms, multiplying a single term over a bracket, taking out common factors

1 · Same value, different shape

Here is the whole page in one line. The three expressions are equal for every value of x; what changes is how they're written and what each form is good for.

Factorised3(x + 2)
expand →
← factorise
Expanded3x + 6

Check it by substituting. Move x and all three forms give the same number.

2 · Simplifying: collect like terms

Like terms have exactly the same letter part: 3x and 5x are like; 3x and 3x² are not; 3x and 3y are not. Only like terms combine, because only tiles of the same shape stack. Type an expression and watch the tiles sort themselves into groups.

The sign belongs to the term. In 4x + 3 − x, the x-terms are +4x and −x, giving 3x. Read each term with the sign in front of it, and a lot of errors vanish. Also: 3x + 2 cannot become 5x. You don't know how many unit squares make a strip, so they stay separate.

3 · Expanding a single bracket

3(x + 2) means three lots of (x + 2). Picture a rectangle 3 high and (x + 2) wide. Its area is the whole thing, but you can also split it into a 3-by-x part and a 3-by-2 part. Same rectangle, so 3(x + 2) = 3x + 6. The number outside multiplies every term inside. Drag the sliders to change the rectangle.

This picture is the grid method you used for long multiplication, with a letter in one of the columns. 3 × 12 = 3 × (10 + 2) = 30 + 6 and 3(x + 2) = 3x + 6 are the same calculation.

With a letter outside

x(x + 4) works the same way: an x-by-x square and an x-by-4 strip. The square is x², so x(x + 4) = x² + 4x. Negative numbers follow the sign rules: −2(x − 5) = −2x + 10, because (−2) × (−5) is +10.

The classic mistake. 3(x + 2) = 3x + 2. The 3 only reached the first term. Think of the rectangle: leaving out the 3 × 2 corner means the area is wrong. Every term inside gets multiplied, every time.

4 · Factorising: find the sides

Factorising is expanding in reverse. You're given the area, 6x + 9, and asked for the sides of the rectangle. Look for the biggest thing that divides into every term: the highest common factor (see prime factorisation). Here it's 3. Pull it outside, and what's left inside is what you'd multiply 3 by to get each term back.

A reliable routine:

  1. Find the HCF of the numbers.
  2. Find the highest power of each letter that appears in every term.
  3. Write the HCF outside the bracket.
  4. Divide each term by it to fill the bracket.
  5. Expand to check. You should get back exactly what you started with.

"Fully factorise" means the HCF, not just any factor. 4x + 8 = 2(2x + 4) is true but not fully factorised, because 2x + 4 still has a common factor of 2. The answer is 4(x + 2). The check: the bracket should have no common factor left in it. Also, if the first term is negative, take the minus out too: −6x − 9 = −3(2x + 3).

5 · The round trip

Type any expression of the form a(bx + c) or ax + b and the tool goes both ways, checking one against the other. If the two columns don't match, something's wrong.

Expanded
Factorised

6 · Where these show up

Mental arithmetic

7 × 98 = 7(100 − 2) = 700 − 14 = 686. Expanding a bracket is how you multiply awkward numbers in your head. 23 × 5 + 23 × 5 = 23(5 + 5) = 230 is factorising.

Perimeters and areas

A rectangle 2x + 3 by x has perimeter 2(2x + 3) + 2x = 6x + 6 = 6(x + 1), and area x(2x + 3) = 2x² + 3x. Both moves in one question.

Solving equations

3(x − 4) = 15 is easier once expanded: 3x − 12 = 15. Next year's quadratics are solved by factorising: x² + 5x = 0 → x(x + 5) = 0 → x = 0 or −5.

Programming

Compilers factorise: a*b + a*c becomes a*(b+c), one multiplication instead of two. Spreadsheet formulas simplify the same way.

Discounts

Price p with 20% off then £5 off is 0.8p − 5. Three items: 3(0.8p − 5) = 2.4p − 15. Expanding gives you the total straight from p.

Proof

"The sum of three consecutive numbers is a multiple of 3": n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). Simplify, then factorise, and the 3 outside is the proof.

Worked examples

Simplify 7a − 3b + 2a + 5b − a
a-terms: +7a + 2a − a = 8a.
b-terms: −3b + 5b = 2b.
Answer: 8a + 2b. (Different letters stay apart.)
Expand and simplify 4(2x + 1) − 3(x − 2)
First bracket: 4 × 2x = 8x, 4 × 1 = 4 → 8x + 4.
Second bracket, with the minus: −3 × x = −3x, −3 × (−2) = +6 → −3x + 6.
Collect: 8x − 3x = 5x; 4 + 6 = 10. 5x + 10. (Or 5(x + 2) if asked to factorise.)
Fully factorise 12x² − 18x
Numbers: HCF(12, 18) = 6.
Letters: x² and x both contain x, so x comes out. HCF = 6x.
Divide each term: 12x² ÷ 6x = 2x; 18x ÷ 6x = 3. So 6x(2x − 3).
Check: 6x × 2x = 12x², 6x × (−3) = −18x ✓
Exam-style: show that 5(n + 2) − 3(n − 1) is always odd for whole numbers n
Expand: 5n + 10 − 3n + 3.
Simplify: 2n + 13.
Reason: 2n is even for any whole number n (it's 2 × something), and even + 13 is odd. Done. The algebra turned "always" into one line.

Quick check

What you've learned

Simplifying collects like terms, and only like terms: the sign travels with the term. Expanding multiplies the outside by every term inside, which the rectangle picture makes impossible to forget. Factorising runs the rectangle backwards: find the HCF of numbers and letters, put it outside, divide to fill the bracket, and expand to check. "Fully" means the bracket has nothing left in common. The three skills are one skill with the rectangle viewed from different sides.