Question

expand the expression to a polynomial in standard form:
$(x + 1)^3$

Explanation:

Step1: Recall the binomial expansion formula

The binomial expansion of \((a + b)^n\) is given by \(\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\), where \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\). For \((x + 1)^3\), \(a=x\), \(b = 1\), and \(n = 3\).

Step2: Calculate each term

• When \(k = 0\): \(\binom{3}{0}x^{3-0}(1)^{0}=\frac{3!}{0!(3 - 0)!}x^{3}\times1 = 1\times x^{3}\times1=x^{3}\)
• When \(k = 1\): \(\binom{3}{1}x^{3 - 1}(1)^{1}=\frac{3!}{1!(3 - 1)!}x^{2}\times1=\frac{3\times2!}{1\times2!}x^{2}=3x^{2}\)
• When \(k = 2\): \(\binom{3}{2}x^{3 - 2}(1)^{2}=\frac{3!}{2!(3 - 2)!}x^{1}\times1=\frac{3\times2!}{2!\times1!}x = 3x\)
• When \(k = 3\): \(\binom{3}{3}x^{3 - 3}(1)^{3}=\frac{3!}{3!(3 - 3)!}x^{0}\times1 = 1\times1\times1 = 1\)

Step3: Sum the terms

Add all the terms together: \(x^{3}+3x^{2}+3x + 1\)

Answer:

\(x^{3}+3x^{2}+3x + 1\)