Binomial Expansion Flashcards

All 10 cards in this deck

In the expansion of $(a+b)^n$, what is the term containing $b^r$?

e.g. the term in $b^2$ in $(a+b)^6$

$\binom{n}{r}a^{n-r}b^r$

e.g. $\binom{6}{2}a^4b^2 = 15a^4b^2$

What is $\binom{n}{r}$ (also written $^nC_r$) equal to as a formula?

$\binom{n}{r}=\dfrac{n!}{r!(n-r)!}$

e.g. $\binom{5}{2}=\dfrac{120}{2\times 6}=10$

What are the steps to write the next row of Pascal's triangle?

e.g. the row after $1,\ 4,\ 6,\ 4,\ 1$

1. Start with 1: $1$
2. Add each adjacent pair: $1+4=5$, $4+6=10$, $6+4=10$, $4+1=5$
3. End with 1: row is $1,\ 5,\ 10,\ 10,\ 5,\ 1$ (coefficients for power 5)

What are the steps to expand and simplify a cube of a bracket?

e.g. $(5x+2)^3$

1. Coefficients from Pascal's triangle: $1,\ 3,\ 3,\ 1$
2. Terms: $(5x)^3+3(5x)^2(2)+3(5x)(2)^2+(2)^3$
3. Simplify: $125x^3+150x^2+60x+8$

What happens to the signs when you expand $(a-bx)^n$?

e.g. $(1-x)^4$

The signs alternate $+,-,+,-,\dots$ because the whole $(-bx)$ is raised to each power.

e.g. $1-4x+6x^2-4x^3+x^4$

True or false? In the expansion of $(2+3x)^4$ the $x^2$ term is $\binom{4}{2}\times 2^2\times 3x^2$.

False. The whole $3x$ must be squared: $\binom{4}{2}\times 2^2\times (3x)^2 = 216x^2$.

What are the steps to find the coefficient of a given power of $x$ without expanding fully?

e.g. the coefficient of $x^2$ in $(3+ax)^5$

1. Choose $r$ to match the power: $r=2$
2. Write $\binom{n}{r}\times(\text{first term})^{n-r}\times(\text{second term})^r$: $\binom{5}{2}\times 3^3\times (ax)^2$
3. Evaluate: $10\times 27\times a^2 = 270a^2$

True or false? In the term $270a^2x^2$, the coefficient of $x^2$ is $270a^2$.

True. The coefficient is the whole factor multiplying $x^2$, without the $x^2$ itself.

What are the steps to find an unknown by comparing coefficients in two expansions?

e.g. the coefficient of $x^2$ in $(2+3x)^4$ equals the coefficient of $x^2$ in $(a+x)^3$

1. Coefficient from the first expansion: $\binom{4}{2}\times 2^2\times 3^2=216$
2. Coefficient from the second: $\binom{3}{2}\times a=3a$
3. Equate and solve: $3a=216$, so $a=72$

What are the steps when one coefficient of an expansion is a multiple of another?

e.g. in $(3+ax)^5$ the coefficient of $x^4$ is $8$ times the coefficient of $x^2$

1. Coefficient of $x^2$: $\binom{5}{2}\times 3^3\times a^2=270a^2$
2. Coefficient of $x^4$: $\binom{5}{4}\times 3\times a^4=15a^4$
3. Form and solve the equation: $15a^4=8\times 270a^2$, so $a^2=144$ and $a=\pm 12$