Problem Solving with Binomial Expansion Flashcards

All 4 cards in this deck

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 $a=\pm 12$