Algebra 2 Flashcards: Applying The Binomial Theorem

What this deck covers

This deck focuses on Applying The Binomial Theorem, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find the value of $\binom{9}{4}$.

Answer: $126$. Using $\binom{9}{4}=\frac{9!}{4!5!}=126$.

Flashcard 2: Expand $(x+y)^3$.

Answer: $x^3+3x^2y+3xy^2+y^3$. Using coefficients $1,3,3,1$ from Pascal's triangle.

Flashcard 3: Expand $(x-y)^3$.

Answer: $x^3-3x^2y+3xy^2-y^3$. Alternating signs from $(-y)^k$ with coefficients $1,3,3,1$.

Flashcard 4: What is the coefficient of $x^5y^3$ in $(x-y)^8$?

Answer: $-\binom{8}{3}=-56$. Coefficient is $\binom{8}{3}=56$ with negative sign from $(-y)^3$.

Flashcard 5: Find the value of $\binom{10}{1}$.

Answer: $10$. Using $\binom{10}{1}=10$.

Flashcard 6: What is the factorial definition of the binomial coefficient $\binom{n}{k}$?

Answer: $\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Uses factorial formula to calculate combinations.

Flashcard 7: State the Binomial Theorem formula for expanding $(x+y)^n$ using binomial coefficients.

Answer: $(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k$. The general formula for binomial expansion with coefficients and powers.

Flashcard 8: Expand $(x-y)^3$.

Answer: $x^3-3x^2y+3xy^2-y^3$. Alternating signs from $(-y)^k$ with coefficients $1,3,3,1$.

Flashcard 9: Expand $(x-y)^4$.

Answer: $x^4-4x^3y+6x^2y^2-4xy^3+y^4$. Alternating signs from $(-y)^k$ with coefficients $1,4,6,4,1$.

Flashcard 10: What is the coefficient of $x^7y^2$ in $(x+y)^9$?

Answer: $\binom{9}{2}=36$. The coefficient of $x^7y^2$ is $\binom{9}{2}=36$.

Flashcard 11: What is the coefficient of $x^3y^5$ in $(x+y)^8$?

Answer: $\binom{8}{5}=56$. The coefficient of $x^3y^5$ is $\binom{8}{5}=56$.

Flashcard 12: What is the coefficient of $x^6y^4$ in $(x+y)^{10}$?

Answer: $\binom{10}{4}=210$. The coefficient of $x^6y^4$ is $\binom{10}{4}=210$.

Flashcard 13: What is the coefficient of $x^5y^3$ in $(x-y)^8$?

Answer: $-\binom{8}{3}=-56$. Coefficient is $\binom{8}{3}=56$ with negative sign from $(-y)^3$.

Flashcard 14: What is the coefficient of $x^4y^4$ in $(x-y)^8$?

Answer: $\binom{8}{4}=70$. Coefficient is $\binom{8}{4}=70$ with positive sign from $(-y)^4$.

Flashcard 15: Identify the coefficient of the $x^2y^3$ term in $(x+y)^5$.

Answer: $\binom{5}{3}=10$. The coefficient of $x^2y^3$ is $\binom{5}{3}=10$.

Flashcard 16: What is the coefficient of $a^4b^2$ in $(a+b)^6$?

Answer: $\binom{6}{2}=15$. The coefficient of $a^4b^2$ is $\binom{6}{2}=15$.

Flashcard 17: What is the coefficient of $a^2b^4$ in $(a+b)^6$?

Answer: $\binom{6}{4}=15$. The coefficient of $a^2b^4$ is $\binom{6}{4}=15$.

Flashcard 18: What is the coefficient pattern (Pascal row) for expanding $(x+y)^3$?

Answer: $1,3,3,1$. Row 3 of Pascal's triangle.

Flashcard 19: Find the value of $\binom{8}{2}$.

Answer: $28$. Using $\binom{8}{2}=\frac{8!}{2!6!}=\frac{8 \cdot 7}{2}=28$.

Flashcard 20: Find the value of $\binom{8}{6}$.

Answer: $28$. Using symmetry: $\binom{8}{6}=\binom{8}{2}=28$.

Flashcard 21: Find the value of $\binom{10}{1}$.

Answer: $10$. Using $\binom{10}{1}=10$.

Flashcard 22: Find the value of $\binom{10}{2}$.

Answer: $45$. Using $\binom{10}{2}=\frac{10 \cdot 9}{2}=45$.

Flashcard 23: Find the value of $\binom{10}{3}$.

Answer: $120$. Using $\binom{10}{3}=\frac{10 \cdot 9 \cdot 8}{6}=120$.

Flashcard 24: What is the coefficient sum of $(x-y)^n$ (equivalently, evaluate at $x=1,y=1$)?

Answer: $0$ if $n$ is odd; $2^n$ if $n$ is even. Substitute $x=1$ and $y=-1$ into $(x-y)^n$.

Flashcard 25: In $(x+y)^n$, what is the general term (the $k$th term) written in powers of $x$ and $y$?

Answer: $\binom{n}{k}x^{n-k}y^k$. The $(k+1)$th term in the binomial expansion.

Flashcard 26: Find the value of $\binom{9}{4}$.

Answer: $126$. Using $\binom{9}{4}=\frac{9!}{4!5!}=126$.

Flashcard 27: Expand $(x+y)^2$.

Answer: $x^2+2xy+y^2$. Using coefficients $1,2,1$ from Pascal's triangle.

Flashcard 28: Expand $(x+y)^3$.

Answer: $x^3+3x^2y+3xy^2+y^3$. Using coefficients $1,3,3,1$ from Pascal's triangle.

Flashcard 29: Expand $(x+y)^4$.

Answer: $x^4+4x^3y+6x^2y^2+4xy^3+y^4$. Using coefficients $1,4,6,4,1$ from Pascal's triangle.

Flashcard 30: In $(x+y)^n$, what is the exponent of $x$ in the term containing $y^k$?

Answer: $n-k$. Exponents of $x$ and $y$ must sum to $n$.