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: 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 2: What is the coefficient of $x^5y$ in $(x-y)^6$?

Answer: $-\binom{6}{1}=-6$. The coefficient is $\binom{6}{1}=6$ with negative sign from $(-y)^1$.

Flashcard 3: What is the coefficient of $x^3$ in $(x+2)^5$?

Answer: $\binom{5}{2}2^2=40$. From term $\binom{5}{2}x^3(2)^2=10 \cdot 4x^3=40x^3$.

Flashcard 4: Find the two middle terms of $(x+y)^7$.

Answer: $35x^4y^3$ and $35x^3y^4$. For odd $n=7$, terms 4 and 5 are both middle terms.

Flashcard 5: Find the value of $\binom{8}{2}$.

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

Flashcard 6: 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 7: What are the values of $\binom{n}{0}$ and $\binom{n}{n}$ for any positive integer $n$?

Answer: $\binom{n}{0}=1$ and $\binom{n}{n}=1$. Only one way to choose all or none of the elements.

Flashcard 8: Identify the term containing $x^3$ in the expansion of $(x+y)^5$.

Answer: $\binom{5}{2}x^3y^2=10x^3y^2$. The term with $y^2$ gives $\binom{5}{2}x^3y^2=10x^3y^2$.

Flashcard 9: 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 10: 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 11: 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 $n$.

Flashcard 12: 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 13: Find the value of $\binom{9}{4}$.

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

Flashcard 14: Expand $(x+y)^3$.

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

Flashcard 15: Find the value of $\binom{10}{3}$.

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

Flashcard 16: 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 17: What is the coefficient of $x^2$ in $(3x-2)^5$?

Answer: $\binom{5}{2}3^2(-2)^3=-720$. From term $\binom{5}{2}(3x)^2(-2)^3=10 \cdot 9 \cdot (-8)x^2=-720x^2$.

Flashcard 18: What is the constant term of $(x-2)^7$?

Answer: $(-2)^7=-128$. The last term with $x^0$ gives $(-2)^7=-128$.

Flashcard 19: What is the coefficient pattern (Pascal row) for expanding $(x+y)^6$?

Answer: $1,6,15,20,15,6,1$. Row 6 of Pascal's triangle.

Flashcard 20: Expand $(x+y)^2$.

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

Flashcard 21: What is the coefficient of $x^4$ in $(1-2x)^6$?

Answer: $\binom{6}{4}(-2)^4=240$. From term $\binom{6}{4}(1)^2(-2x)^4=15 \cdot 16x^4=240x^4$.

Flashcard 22: 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 23: 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 24: How many terms are in the expanded form of $(x+y)^n$ when like terms are combined?

Answer: $n+1$. Terms range from $k=0$ to $k=n$.

Flashcard 25: What is the coefficient of $x^5$ in $(2x+1)^7$?

Answer: $\binom{7}{5}2^5=672$. From term $\binom{7}{5}(2x)^5(1)^2=21 \cdot 32x^5=672x^5$.

Flashcard 26: Find the value of $\binom{10}{1}$.

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

Flashcard 27: What is the constant term of $(2x-5)^6$?

Answer: $(-5)^6=15625$. The last term with $(2x)^0$ gives $(-5)^6=15625$.

Flashcard 28: Identify the term containing $y^4$ in the expansion of $(x+y)^7$.

Answer: $\binom{7}{4}x^3y^4=35x^3y^4$. The term with $x^3$ gives $\binom{7}{4}x^3y^4=35x^3y^4$.

Flashcard 29: 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 30: Find the middle term of $(x+y)^8$.

Answer: $\binom{8}{4}x^4y^4=70x^4y^4$. For even $n=8$, the middle term is $\binom{8}{4}x^4y^4=70x^4y^4$.

Flashcard 31: What is the coefficient of $x^4$ in $(x+2)^6$?

Answer: $\binom{6}{2}2^2=60$. From term $\binom{6}{2}x^4(2)^2=15 \cdot 4x^4=60x^4$.

Flashcard 32: What is the coefficient of $x^2$ in $(x-3)^4$?

Answer: $\binom{4}{2}(-3)^2=54$. From term $\binom{4}{2}x^2(-3)^2=6 \cdot 9x^2=54x^2$.

Flashcard 33: Find the two middle terms of $(x + y)^7$.

Answer: $35x^4y^3$ and $35x^3y^4$. For odd $n=7$, terms 4 and 5 are both middle terms.

Flashcard 34: 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 35: Identify the sign pattern of terms in $(x-y)^n$ as $k$ increases in $\binom{n}{k}x^{n-k}(-y)^k$.

Answer: Signs alternate by $(-1)^k$. The factor $(-1)^k$ creates alternating positive and negative terms.

Flashcard 36: What is the symmetry identity for binomial coefficients relating $\binom{n}{k}$ and $\binom{n}{n-k}$?

Answer: $\binom{n}{k}=\binom{n}{n-k}$. Choosing $k$ from $n$ equals choosing $n-k$ from $n$.

Flashcard 37: What is the coefficient of $x^2y^4$ in $(x-y)^6$?

Answer: $\binom{6}{4}=15$. The coefficient is $\binom{6}{4}=15$ with positive sign from $(-y)^4$.

Flashcard 38: 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 39: 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 40: Find the value of $\binom{8}{6}$.

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

Flashcard 41: What is the coefficient pattern (Pascal row) for expanding $(x+y)^3$?

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

Flashcard 42: 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 43: What is the coefficient pattern (Pascal row) for expanding $(x+y)^5$?

Answer: $1,5,10,10,5,1$. Row 5 of Pascal's triangle.

Flashcard 44: In $(x+y)^n$, what is the exponent of $y$ in the term containing $x^{n-k}$?

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

Flashcard 45: 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 46: Find the value of $\binom{10}{2}$.

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

Flashcard 47: 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 48: What is the sum of the coefficients of $(x+y)^n$ (equivalently, evaluate at $x=1,y=1$)?

Answer: $2^n$. Substitute $x=1$ and $y=1$ into $(x+y)^n$.

Flashcard 49: What is the coefficient of $x^3y^3$ in $(x+y)^6$?

Answer: $\binom{6}{3}=20$. The coefficient of $x^3y^3$ is $\binom{6}{3}=20$.

Flashcard 50: What is the coefficient pattern (Pascal row) for expanding $(x+y)^4$?

Answer: $1,4,6,4,1$. Row 4 of Pascal's triangle.

Flashcard 51: What is Pascal's identity for binomial coefficients?

Answer: $\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$. Each coefficient equals the sum of two above it in Pascal's triangle.