Question

name aubrey kirpela
use pascals triangle and binomial theorem.
⑨ ((x + 4)^5)
⑩ ((3x + 4y)^4)
⑪ ((5p + 6q)^7)

Explanation:

Problem 9: \((x + 4)^5\)

Step 1: Recall Pascal's Triangle for \(n = 5\)

The 5th row (starting from row 0) of Pascal's Triangle is \(1, 5, 10, 10, 5, 1\). The binomial theorem states that \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\), where \(\binom{n}{k}\) are the coefficients from Pascal's Triangle. Here, \(a=x\), \(b = 4\), and \(n = 5\).

Step 2: Apply the Binomial Theorem

$$
LATEXBLOCK0
$$

Problem 10: \((3x + 4y)^4\)

Step 1: Recall Pascal's Triangle for \(n=4\)

The 4th row of Pascal's Triangle is \(1, 4, 6, 4, 1\). Using the binomial theorem \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\), here \(a = 3x\), \(b=4y\), and \(n = 4\).

Step 2: Apply the Binomial Theorem

$$
LATEXBLOCK0
$$

Problem 11: \((5p+6q)^{7}\)

Step 1: Recall Pascal's Triangle for \(n = 7\)

The 7th row of Pascal's Triangle is \(1, 7, 21, 35, 35, 21, 7, 1\). Using the binomial theorem \((a + b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n - k}b^{k}\), here \(a = 5p\), \(b = 6q\), and \(n=7\).

Step 2: Apply the Binomial Theorem

$$
LATEXBLOCK0
$$

Answer:

s:

• \((x + 4)^5=x^{5}+20x^{4}+160x^{3}+640x^{2}+1280x + 1024\)
• \((3x + 4y)^4=81x^{4}+432x^{3}y + 864x^{2}y^{2}+768xy^{3}+256y^{4}\)
• \((5p + 6q)^{7}=78125p^{7}+656250p^{6}q + 2205000p^{5}q^{2}+4725000p^{4}q^{3}+5904900p^{3}q^{4}+4082400p^{2}q^{5}+1633960pq^{6}+279936q^{7}\)