Expanding Brackets Flashcards

All 17 cards in this deck

What does it mean to expand a single bracket?

e.g. $3(x+4)$

Multiply every term inside the bracket by the term outside.
$3(x+4)=3x+12$

True or false? $4(x+3)$ expands to $4x+3$.

False. The $4$ multiplies both terms: $4x+12$.

How do you expand a bracket when the term outside contains a variable?

e.g. $4p(p^2+3p)$

Multiply each term inside by the whole term outside, adding the indices.
$4p(p^2+3p)=4p^3+12p^2$

What are the steps to expand and simplify two single brackets with a subtraction in front of the second?

e.g. $5(p+3)-2(1-2p)$

1. Expand the first bracket: $5p+15$
2. Multiply the second bracket by $-2$ throughout: $-2+4p$
3. Collect like terms: $9p+13$

True or false? $-2(1-2p)$ expands to $-2-4p$.

False. It is $-2+4p$, because $-2\times-2p=+4p$.

What happens to the terms when a bracket is subtracted?

e.g. $-(x^2-2x-4)$

Every term inside changes sign.
$-(x^2-2x-4)=-x^2+2x+4$

What are the steps to expand and simplify the product of two binomials?

e.g. $(x+5)(x-9)$

1. Multiply each term in the first bracket by each term in the second: $x^2-9x+5x-45$
2. Collect the like terms: $x^2-4x-45$

How many terms do you get when expanding two binomials, before simplifying?

Four terms.
e.g. $(x+5)(x-9)=x^2-9x+5x-45$

In the expansion of $(x+a)(x+b)$, what are the coefficient of $x$ and the constant term?

Coefficient of $x$ is $a+b$; constant term is $ab$.
e.g. $(x+2)(x+5)=x^2+7x+10$

How do you expand a squared bracket?

e.g. $(2x-5)^2$

Write it as two brackets and expand.
$(2x-5)(2x-5)=4x^2-20x+25$

True or false? $(x+3)^2=x^2+9$.

False. $(x+3)^2=x^2+6x+9$ — the middle terms must be included.

What are the steps to expand and simplify three binomials?

e.g. $(x+1)(x+2)(x+3)$

1. Multiply two of the brackets: $(x+1)(x+2)=x^2+3x+2$
2. Multiply that quadratic by the remaining bracket, each term by each: $x^3+3x^2+2x+3x^2+9x+6$
3. Collect like terms: $x^3+6x^2+11x+6$

What form does the expansion of three linear brackets take?

A cubic, $ax^3+bx^2+cx+d$.

How many terms do you get when multiplying a simplified 3-term quadratic by a binomial, before simplifying?

Six terms.

True or false? When expanding three brackets you must multiply the first two brackets first.

False. Any pair can be multiplied first; the final cubic is the same.

When three linear brackets are expanded to $ax^3+bx^2+cx+d$, how can you check $a$ and $d$ quickly?

e.g. $(2x+1)(x+3)(3x+7)$

$a$ is the product of the $x$-coefficients and $d$ is the product of the constants.
$a=2\times1\times3=6$, $d=1\times3\times7=21$

True or false? $(x-2)(x+5)$ and $(x+5)(x-2)$ give the same expansion.

True. Multiplication is commutative, so the order of the two brackets does not change the result.