Expanding Brackets flashcards

Key points

• Multiply every term inside the bracket by the term outside.
• Keep each result's sign: 3(2x + 5) = 6x + 15.
• A letter outside multiplies too: x(x + 4) = x² + 4x.

7(3x − 4)

Key points

• The negative multiplies every term inside, so every sign inside is affected.
• Negative × positive = negative; negative × negative = positive.
• So −2(x − 4) = −2x + 8.

Expand −4(3x − 5)

Key points

• An expression has no equals sign: 4x − 7.
• An equation is true only for some values of x, so it can be solved.
• An identity is true for every value of the letters, and uses ≡: 2(x + 3) ≡ 2x + 6.
• Expanding a bracket always gives an identity.

Show 3(x + 4) − 2x ≡ x + 12

Key points

• Expand each bracket on its own first, each on its own line if that helps.
• Keep the sign in front of a bracket with it: − 2(x − 5) multiplies by −2.
• Then collect like terms: the x terms together, the numbers together.

3(2x + 4) − 2(x − 5)

Key points

• Like terms have exactly the same letters and powers: 3x and 5x.
• x and x² are not like terms, so 3x + 5x² stays as it is.
• The coefficient is the number in front of the letter: 7 in 7y. On its own, y means 1y.

Simplify 4x² + 3x − x² + 5x − 2

Key points

• Every term in the first bracket multiplies every term in the second: four products.
• FOIL keeps track: First, Outer, Inner, Last.
• The two middle terms are like terms: collect them.

(x + 6)(x − 4)

Key points

• Write it as the bracket times itself: (x + 3)(x + 3).
• Expand all four products: there is always a middle term.
• (x + a)² = x² + 2ax + a², so (x + 3)² = x² + 6x + 9.

(x − 7)²

Key points

• Exactly as before: every term times every term, four products.
• Multiply the numbers and the letters: 2x × 3x = 6x².
• Collect the two middle terms.

(2x + 3)(3x − 1)

Key points

• Factorising is the reverse of expanding: it writes the expression as a product.
• Find the highest common factor of every term, numbers and letters.
• Put it outside the bracket; inside goes each term divided by it.
• Check by expanding your answer back out.

Factorise 12x² + 18x

Key points

• Find two numbers that multiply to c and add to b.
• If c is positive, both numbers have the sign of b; if c is negative, their signs differ.
• Write them in (x + p)(x + q), then expand to check.

x² − 3x − 10 into two brackets

Key points

• One square is taken away from another, and there is no middle term.
• x² − a² = (x + a)(x − a).
• It works because the middle terms, +ax and −ax, cancel when you expand.

Example

Key points

• Multiply two of the brackets together first.
• Then multiply every term of that result by the third bracket: six products.
• Collect like terms; a grid keeps every product in view.

(x + 2)(x − 1)(x + 3)