Question

question 4 (essay worth 4 points) (h1.01h mc) expand (x - 2)^5 using the binomial theorem and pascals triangle. show all necessary steps.

Explanation:

Step1: Recall Pascal's Triangle for n=5

Pascal's Triangle for the 5th power (n=5) has coefficients: 1, 5, 10, 10, 5, 1.

Step2: Apply Binomial Theorem

The Binomial Theorem states \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\). For \((x-2)^{5}\), let \(a=x\), \(b = - 2\), \(n = 5\).
Using the coefficients from Pascal's Triangle:

• Term 1 (\(k = 0\)): \(1\times x^{5}\times(-2)^{0}=x^{5}\)
• Term 2 (\(k = 1\)): \(5\times x^{4}\times(-2)^{1}=5x^{4}\times(-2)=-10x^{4}\)
• Term 3 (\(k = 2\)): \(10\times x^{3}\times(-2)^{2}=10x^{3}\times4 = 40x^{3}\)
• Term 4 (\(k = 3\)): \(10\times x^{2}\times(-2)^{3}=10x^{2}\times(-8)=-80x^{2}\)
• Term 5 (\(k = 4\)): \(5\times x^{1}\times(-2)^{4}=5x\times16 = 80x\)
• Term 6 (\(k = 5\)): \(1\times x^{0}\times(-2)^{5}=1\times1\times(-32)=-32\)

Step3: Combine Terms

Add all the terms together: \(x^{5}-10x^{4}+40x^{3}-80x^{2}+80x - 32\)

Answer:

\(x^{5}-10x^{4}+40x^{3}-80x^{2}+80x - 32\)