Question

expand and state your answer as a polynomial in standard form.
$(2x^2 + y^4)^2$

Explanation:

Step1: Apply the square of a binomial formula \((a + b)^2 = a^2 + 2ab + b^2\)

Here, \(a = 2x^2\) and \(b = y^4\). So we have \((2x^2)^2 + 2\times(2x^2)\times(y^4) + (y^4)^2\)

Step2: Simplify each term

• For \((2x^2)^2\), using the power of a product rule \((ab)^n = a^n b^n\) and \((a^m)^n = a^{mn}\), we get \(2^2\times(x^2)^2 = 4x^4\)
• For \(2\times(2x^2)\times(y^4)\), we multiply the coefficients and keep the variables as they are, so we get \(4x^2y^4\)
• For \((y^4)^2\), using the power of a power rule \((a^m)^n = a^{mn}\), we get \(y^{4\times2} = y^8\)

Step3: Combine the simplified terms

Putting them together, we have \(4x^4 + 4x^2y^4 + y^8\)

Answer:

\(4x^4 + 4x^2y^4 + y^8\)