expand the function. f(x)=(x - 3)^4 ?x^4+ x^3+ x^2+ x+
expand the function. f(x)=(x - 3)^4 ?x^4+ x^3+ x^2+ x+
Answer
Explanation:
Step1: Use binomial theorem
The binomial theorem states that $(a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}$. Here $a=x$, $b=- 3$, and $n = 4$.
Step2: Calculate binomial coefficients and powers
$\binom{4}{0}x^{4}(-3)^{0}=1\times x^{4}\times1=x^{4}$ $\binom{4}{1}x^{3}(-3)^{1}=\frac{4!}{1!(4 - 1)!}\times x^{3}\times(-3)=4\times x^{3}\times(-3)=-12x^{3}$ $\binom{4}{2}x^{2}(-3)^{2}=\frac{4!}{2!(4 - 2)!}\times x^{2}\times9 = 6\times x^{2}\times9 = 54x^{2}$ $\binom{4}{3}x^{1}(-3)^{3}=\frac{4!}{3!(4 - 3)!}\times x\times(-27)=4\times x\times(-27)=-108x$ $\binom{4}{4}x^{0}(-3)^{4}=1\times1\times81 = 81$
Answer:
$1x^{4}-12x^{3}+54x^{2}-108x + 81$