Question

1. expand using the binomial theorem.

$(3x + 5)^4$

Explanation:

Step1: Recall binomial theorem

The binomial theorem for $(a+b)^n$ is $\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}$, where $\binom{n}{k}=\frac{n!}{k!(n-k)!}$

Step2: Identify $a, b, n$

Here, $a=3x$, $b=5$, $n=4$

Step3: Calculate each term for $k=0$

Compute first term: $\binom{4}{0}(3x)^4(5)^0$
$\binom{4}{0}=1$, $(3x)^4=81x^4$, $(5)^0=1$
Term: $1 \cdot 81x^4 \cdot 1 = 81x^4$

Step4: Calculate term for $k=1$

Compute second term: $\binom{4}{1}(3x)^3(5)^1$
$\binom{4}{1}=4$, $(3x)^3=27x^3$, $(5)^1=5$
Term: $4 \cdot 27x^3 \cdot 5 = 540x^3$

Step5: Calculate term for $k=2$

Compute third term: $\binom{4}{2}(3x)^2(5)^2$
$\binom{4}{2}=6$, $(3x)^2=9x^2$, $(5)^2=25$
Term: $6 \cdot 9x^2 \cdot 25 = 1350x^2$

Step6: Calculate term for $k=3$

Compute fourth term: $\binom{4}{3}(3x)^1(5)^3$
$\binom{4}{3}=4$, $(3x)^1=3x$, $(5)^3=125$
Term: $4 \cdot 3x \cdot 125 = 1500x$

Step7: Calculate term for $k=4$

Compute fifth term: $\binom{4}{4}(3x)^0(5)^4$
$\binom{4}{4}=1$, $(3x)^0=1$, $(5)^4=625$
Term: $1 \cdot 1 \cdot 625 = 625$

Step8: Sum all terms

Add all computed terms together.

Answer:

$81x^4 + 540x^3 + 1350x^2 + 1500x + 625$