Question

polynomial models

1. verify each algebraic statement by transforming one side of the equation to show that it is equivalent to the other side of the equation.
2. ( g^6 - h^6 = (g^2 - h^2)(g^2 - gh + h^2)(g^2 + gh + h^2) )

1. ( (m^2 + n^2)^3 = (m^2 + n^2)(m^4 + 2m^2n^2 + n^4) )

Explanation:

Problem 1:

Step 1: Recognize the difference of squares and sum/difference of cubes

We know that \(a^3 - b^3=(a - b)(a^2+ab + b^2)\) and \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\), also \(a^6 - b^6=(a^3)^2-(b^3)^2=(a^3 - b^3)(a^3 + b^3)\)
First, factor \(g^6 - h^6\) as a difference of squares: \(g^6 - h^6=(g^3)^2-(h^3)^2=(g^3 - h^3)(g^3 + h^3)\)

Step 2: Factor the sum and difference of cubes

Factor \(g^3 - h^3\) using the difference of cubes formula: \(g^3 - h^3=(g - h)(g^2+gh + h^2)\)
Factor \(g^3 + h^3\) using the sum of cubes formula: \(g^3 + h^3=(g + h)(g^2 - gh + h^2)\)
But also, we can factor \(g^6 - h^6\) as \((g^2)^3-(h^2)^3\)
Using the difference of cubes formula \(a^3 - b^3=(a - b)(a^2+ab + b^2)\) where \(a = g^2\) and \(b=h^2\)
\(g^6 - h^6=(g^2 - h^2)(g^4+g^2h^2 + h^4)\)
Now, factor \(g^2 - h^2=(g - h)(g + h)\) and we can rewrite \(g^4+g^2h^2 + h^4\) as \((g^4 + 2g^2h^2+h^4)-g^2h^2=(g^2 + h^2)^2-(gh)^2=(g^2+gh + h^2)(g^2 - gh + h^2)\)
So \(g^6 - h^6=(g^2 - h^2)(g^2+gh + h^2)(g^2 - gh + h^2)\) and since \(g^2 - h^2=(g - h)(g + h)\), but the right - hand side of the given equation is \((g^2 - h^2)(g^2 - gh + h^2)(g^2+gh + h^2)\), which is equivalent to the factored form of \(g^6 - h^6\)

Problem 2:

Step 1: Recall the formula for \((a + b)^3\)

We know that \((a + b)^3=a^3 + 3a^2b+3ab^2 + b^3\)
Let \(a = m^2\) and \(b = n^2\), then \((m^2 + n^2)^3=(m^2)^3+3(m^2)^2n^2 + 3m^2(n^2)^2+(n^2)^3=m^6+3m^4n^2 + 3m^2n^4 + n^6\)

Step 2: Multiply the right - hand side

Multiply \((m^2 + n^2)(m^4+2m^2n^2 + n^4)\)
Using the distributive property (FOIL method for polynomials):
\((m^2)(m^4)+(m^2)(2m^2n^2)+(m^2)(n^4)+n^2(m^4)+n^2(2m^2n^2)+n^2(n^4)\)
\(=m^6+2m^4n^2+m^2n^4+m^4n^2 + 2m^2n^4+n^6\)
Combine like terms:
\(m^6+(2m^4n^2+m^4n^2)+(m^2n^4 + 2m^2n^4)+n^6=m^6+3m^4n^2+3m^2n^4 + n^6\)
Which is equal to \((m^2 + n^2)^3\)
Also, we can use the formula \(a^3=(a)(a^2)\), here \((m^2 + n^2)^3=(m^2 + n^2)\times(m^2 + n^2)^2\)
And \((m^2 + n^2)^2=m^4 + 2m^2n^2 + n^4\)
So \((m^2 + n^2)^3=(m^2 + n^2)(m^4 + 2m^2n^2 + n^4)\)

For problem 1:

Answer:

The equation \(g^6 - h^6=(g^2 - h^2)(g^2 - gh + h^2)(g^2+gh + h^2)\) is verified.

For problem 2: