Applying the Binomial Theorem - Algebra 2

1. What is the coefficient of $x^6y^4$ in $(x+y)^{10}$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

10. Expand $(x-y)^3$.

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

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

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

12. What is the factorial definition of the binomial coefficient $\binom{n}{k}$?

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

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

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

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

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

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

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

16. Expand $(x+y)^4$.

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

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

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

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

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

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

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

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

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

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

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

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

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

23. What is the coefficient of $x^4y^4$ in $(x-y)^8$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

34. Expand $(x-y)^4$.

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

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

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

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

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

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

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

38. What is the coefficient of $a^2b^4$ in $(a+b)^6$?

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

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

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

40. Find the value of $\binom{8}{6}$.

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

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

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

42. What is the coefficient of $a^4b^2$ in $(a+b)^6$?

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

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

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

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

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

45. What is the coefficient of $x^3y^5$ in $(x+y)^8$?

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

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

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

47. What is the coefficient of $x^5y^3$ in $(x-y)^8$?

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

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

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

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

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

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

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

51. What is Pascal's identity for binomial coefficients?

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

Study these flashcards