Math Problem Statement

Expand and state your answer as a polynomial in standard form. left parenthesis, 4, x, cubed, plus, y, cubed, right parenthesis, squared (4x 3 +y 3 ) 2

Solution

We are tasked with expanding and writing the polynomial (4x3+y3)2(4x^3 + y^3)^2 in standard form.

To solve this, we can apply the square of a binomial formula:

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

In this case, a=4x3a = 4x^3 and b=y3b = y^3. Applying the formula:

(4x3+y3)2=(4x3)2+2(4x3)(y3)+(y3)2(4x^3 + y^3)^2 = (4x^3)^2 + 2(4x^3)(y^3) + (y^3)^2

Now, calculate each term:

  1. (4x3)2=16x6(4x^3)^2 = 16x^6
  2. 2(4x3)(y3)=8x3y32(4x^3)(y^3) = 8x^3y^3
  3. (y3)2=y6(y^3)^2 = y^6

Putting it all together:

(4x3+y3)2=16x6+8x3y3+y6(4x^3 + y^3)^2 = 16x^6 + 8x^3y^3 + y^6

This is the expanded polynomial in standard form.


Would you like more details on any step or further clarification?

Here are five related questions:

  1. What is the square of (2x2+3y2)2(2x^2 + 3y^2)^2?
  2. How does the binomial expansion formula apply to cubes?
  3. Can you expand (x2+y2)3(x^2 + y^2)^3 using a similar approach?
  4. What is the general formula for expanding (a+b)n(a + b)^n for any nn?
  5. How do you simplify higher-degree polynomials in standard form?

Tip: Always simplify the individual terms of your expansion before combining them into the final polynomial.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Binomial Expansion

Formulas

(a + b)^2 = a^2 + 2ab + b^2

Theorems

Binomial Expansion Theorem

Suitable Grade Level

Grades 9-12