Question

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

Explanation:

Step1: Write out the coefficients from Pascal's Triangle for power 4

The coefficients for $(a + b)^4$ from Pascal's Triangle are $1,4,6,4,1$.

Step2: Let $a = 2$ and $b=-5x^{2}$

Use the binomial expansion formula $(a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}$. Here $n = 4$.
For $k = 0$:
$$\binom{4}{0}(2)^{4-0}(-5x^{2})^{0}=1\times16\times1 = 16$$
For $k = 1$:
$$\binom{4}{1}(2)^{4 - 1}(-5x^{2})^{1}=4\times8\times(-5x^{2})=-160x^{2}$$
For $k = 2$:
$$\binom{4}{2}(2)^{4-2}(-5x^{2})^{2}=\frac{4!}{2!(4 - 2)!}\times4\times25x^{4}=6\times4\times25x^{4}=600x^{4}$$
For $k = 3$:
$$\binom{4}{3}(2)^{4-3}(-5x^{2})^{3}=4\times2\times(-125x^{6})=-1000x^{6}$$
For $k = 4$:
$$\binom{4}{4}(2)^{4-4}(-5x^{2})^{4}=1\times1\times625x^{8}=625x^{8}$$

Step3: Combine the terms

Add all the terms together: $625x^{8}-1000x^{6}+600x^{4}-160x^{2}+16$.

Answer:

$625x^{8}-1000x^{6}+600x^{4}-160x^{2}+16$