Factorising Flashcards

All 16 cards in this deck

What does the instruction "factorise fully" tell you to do?

Take out the highest common factor of all the terms, so nothing more can be taken out.
e.g. $9x^2 + 6x = 3x(3x + 2)$, not $3(3x^2 + 2x)$

What are the steps to factorise fully an expression with a common factor?

e.g. $15w^2 - 5w$

1. Find the HCF of the numbers: $5$
2. Find the highest power of the letter in every term: $w$, so the HCF is $5w$
3. Divide each term by it and bracket: $5w(3w - 1)$

True or false? $2(2a + 12)$ is a fully factorised form of $4a + 24$.

False.
The HCF is $4$, so the fully factorised answer is $4(a + 6)$.

How can you check that you have factorised an expression correctly?

Expand the brackets again — you should get back the original expression.

True or false? An expression like $x^2 + 7x$, with no number term, can always be factorised by taking out $x$.

True.
Every term contains $x$, so $x^2 + 7x = x(x + 7)$.

To factorise $x^2 + bx + c$ into $(x + a)(x + d)$, what must the two numbers do?

Multiply to give $c$ and add to give $b$.
e.g. $x^2 + 4x + 3 = (x + 1)(x + 3)$

What are the steps to factorise a quadratic of the form $x^2 + bx + c$?

e.g. $x^2 + 10x + 9$

1. List pairs multiplying to $c$: $1 \times 9$, $3 \times 3$
2. Choose the pair adding to $b = 10$: $1$ and $9$
3. Write two brackets: $(x + 1)(x + 9)$

In $x^2 + bx + c$, if $c$ is negative, what signs go in the two brackets?

One $+$ and one $-$.
e.g. $x^2 + 2x - 8 = (x + 4)(x - 2)$

In $x^2 + bx + c$, if $c$ is positive and $b$ is negative, what signs go in the two brackets?

Both $-$.
e.g. $x^2 - 7x + 12 = (x - 3)(x - 4)$

How is the factorised answer written when a quadratic is a perfect square?

e.g. $x^2 + 6x + 9$

As a squared bracket: $(x + 3)^2$, or $(x + 3)(x + 3)$ — both are accepted.

True or false? To factorise $x^2 + bx + c$ you need two numbers that add to $c$ and multiply to $b$.

False.
It is the other way round: they multiply to $c$ and add to $b$.

What is meant by "the difference of two squares"?

An expression of the form $a^2 - b^2$ — one square subtracted from another.
e.g. $x^2 - 16$

What is the factorised form of $x^2 - a^2$?

e.g. $x^2 - 16$

$(x + a)(x - a)$
e.g. $x^2 - 16 = (x + 4)(x - 4)$

What are the steps to factorise a difference of two squares?

e.g. $x^2 - 49$

1. Check it is a square subtract a square: $x^2$ and $7^2$
2. Square root each term: $x$ and $7$
3. Write one $+$ bracket and one $-$ bracket: $(x + 7)(x - 7)$

True or false? $x^2 + 25$ factorises to $(x + 5)(x + 5)$.

False.
A sum of two squares does not factorise; only $x^2 - 25$ does, giving $(x + 5)(x - 5)$.

True or false? $x^2 - 36$ can be factorised even though it has no $x$ term.

True.
It is a difference of two squares: $(x + 6)(x - 6)$.