How is a grid used to expand a pair of double brackets?

Write the terms of one bracket as the column headings and the terms of the other as the row headings, keeping each term's sign attached to it. Multiply each row heading by each column heading to fill in the grid, then add all four entries and collect like terms.

National 5MathsSQAFlashcardsAlgebraic SkillsExpansion of Brackets

Expansion of Brackets (SQA National 5 Maths): Flashcards

Exam code: X847 75

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In the expression $3 x (x + 2)$ what does it mean to expand the bracket?

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In the expression $3 x (x + 2)$ what does it mean to expand the bracket?
Expanding means multiplying the term outside the bracket by each term inside it, which removes the brackets.
So $3 x (x + 2) = 3 x \times x + 3 x \times 2 = 3 x^{2} + 6 x$ .
Complete the expansion by filling in the two missing terms.
$4 x (2 x - 3) =______-______$
The completed expansion is:
$4 x (2 x - 3) = 8 x^{2} - 12 x$
Both terms inside the bracket get multiplied, so expanding a single bracket gives an answer with as many terms as the bracket had.
True or False?
Expanding $- 7 x (4 - 5 y)$ produces a term with a positive sign in it.
True.
Multiplying two negatives gives a positive, so $(- 7 x) \times (- 5 y) = 35 x y$ .
The full expansion is $- 28 x + 35 x y$ , in which only the first term keeps a negative sign.
In the expression $3 x (x + 2) - 7 (x - 6)$ what must the $- 7$ be multiplied by?
The $- 7$ must be multiplied by both terms inside its own bracket, giving $- 7 \times x$ and $- 7 \times (- 6)$ .
That is $- 7 x + 42$ , so the whole expression expands to $3 x^{2} + 6 x - 7 x + 42 = 3 x^{2} - x + 42$ .
What is the first step in expanding and simplifying two brackets added together?
Expand each bracket separately first, and collect like terms only once both have been multiplied out.
For $2 (x + 5) + 3 x (x - 8)$ that gives $2 x + 10 + 3 x^{2} - 24 x$ , which simplifies to $3 x^{2} - 22 x + 10$ .
True or False?
Once two brackets have been expanded, every term in the answer can be collected with another one.
False.
Only like terms can be collected, meaning terms with exactly the same letter raised to exactly the same power.
Expanding $4 (x + 7) + 5 x (3 - x)$ gives $4 x + 28 + 15 x - 5 x^{2}$ , in which only the two $x$ terms combine.
What changes when the term outside a bracket contains a letter as well as a number?
Each multiplication then raises the power of that letter, so a term in $x$ becomes a term in $x^{2}$ .
Expanding $5 (x + 3)$ gives $5 x + 15$ , while expanding $5 x (x + 3)$ gives $5 x^{2} + 15 x$ .
What must be multiplied by what when you expand a pair of double brackets?
Every term in the first bracket must be multiplied by every term in the second.
For $(x + 1) (x + 3)$ that is four multiplications, giving $x^{2} + 3 x + x + 3$ , which simplifies to $x^{2} + 4 x + 3$ .
Complete the expansion by filling in the two missing terms.
$(2 x - 3) (x + 4) = 2 x^{2} +______- 3 x -______$
The completed expansion is:
$(2 x - 3) (x + 4) = 2 x^{2} + 8 x - 3 x - 12$
Collecting the two $x$ terms gives the simplified answer $2 x^{2} + 5 x - 12$ .
How can a pair of double brackets be turned into two single brackets?
Split the terms of the first bracket and multiply the whole second bracket by each of them separately.
So $(x + 1) (x + 3) = x (x + 3) + 1 (x + 3)$ , which is then expanded in the usual way.
True or False?
Expanding a pair of brackets with two terms in each always needs four multiplications.
True.
Each of the two terms in the first bracket has to multiply each of the two terms in the second, which is $2 \times 2 = 4$ products.
Some of those products may combine or cancel afterwards, but they all still have to be found.
How is a grid used to expand a pair of double brackets?
Write the terms of one bracket as the column headings and the terms of the other as the row headings, keeping each term's sign attached to it.
Multiply each row heading by each column heading to fill in the grid, then add all four entries and collect like terms.
Why does expanding $(3 x + 2 y) (4 x - 6 y)$ produce two $x y$ terms?
The outer and the inner multiplications each pair an $x$ with a $y$ , so both of them produce an $x y$ term.
Here they are $- 18 x y$ and $+ 8 x y$ , which combine to give the answer $12 x^{2} - 10 x y - 12 y^{2}$ .
True or False?
${(x + 3)}^{2}$ is equal to $x^{2} + 9$ .
False.
Substituting $x = 1$ settles it, since ${(1 + 3)}^{2} = 16$ but $1^{2} + 3^{2} = 10$ .
Squaring a bracket is not the same as squaring each term inside it, and the correct expansion is $x^{2} + 6 x + 9$ .
When expanding ${(2 x + 3)}^{2}$ what is the first move you should make?
Write the squared bracket out as two separate brackets multiplied together, $(2 x + 3) (2 x + 3)$ , and then expand as normal.
That gives $4 x^{2} + 6 x + 6 x + 9$ , which simplifies to $4 x^{2} + 12 x + 9$ .

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