Question

use pascals triangle to expand $(2 - 5x^{2})^{4}$. express your answer in simplest form.

Explanation:

Step1: Find the coefficients from Pascal's Triangle

For the power of \(4\), the coefficients from Pascal's Triangle are \(1,4,6,4,1\).

Step2: Apply the binomial expansion formula \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\)

Here \(a = 2\), \(b=-5x^{2}\), \(n = 4\)

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Step3: Calculate each term

• For the first term: \(1\times(2)^{4}\times(-5x^{2})^{0}=1\times16\times1 = 16\)
• For the second term: \(4\times(2)^{3}\times(-5x^{2})^{1}=4\times8\times(- 5x^{2})=-160x^{2}\)
• For the third term: \(6\times(2)^{2}\times(-5x^{2})^{2}=6\times4\times25x^{4}=600x^{4}\)
• For the fourth term: \(4\times(2)^{1}\times(-5x^{2})^{3}=4\times2\times(-125x^{6})=-1000x^{6}\)
• For the fifth term: \(1\times(2)^{0}\times(-5x^{2})^{4}=1\times1\times625x^{8}=625x^{8}\)

Step4: Combine all terms

\((2 - 5x^{2})^{4}=625x^{8}-1000x^{6}+600x^{4}-160x^{2}+16\)

Answer:

\(625x^{8}-1000x^{6}+600x^{4}-160x^{2}+16\)