Question

use pascals triangle to expand $(y^{2}+z^{2})^{3}$. express your answer in simplest form.

Explanation:

Step1: Write out Pascal's Triangle row for power 3

Pascal's Triangle row for power \(n = 3\) is \(1,3,3,1\).

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

Here \(a=y^{2}\), \(b = z^{2}\), \(n = 3\).

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Step3: Calculate each term

• For \(k = 0\): \(\binom{3}{0}(y^{2})^{3}(z^{2})^{0}=1\times y^{6}\times1=y^{6}\)
• For \(k = 1\): \(\binom{3}{1}(y^{2})^{2}(z^{2})^{1}=3\times y^{4}\times z^{2}=3y^{4}z^{2}\)
• For \(k = 2\): \(\binom{3}{2}(y^{2})^{1}(z^{2})^{2}=3\times y^{2}\times z^{4}=3y^{2}z^{4}\)
• For \(k = 3\): \(\binom{3}{3}(y^{2})^{0}(z^{2})^{3}=1\times1\times z^{6}=z^{6}\)

Step4: Combine the terms

\((y^{2}+z^{2})^{3}=y^{6}+3y^{4}z^{2}+3y^{2}z^{4}+z^{6}\)

Answer:

\(y^{6}+3y^{4}z^{2}+3y^{2}z^{4}+z^{6}\)