Question

we can factor the expression as ((u - v)^2) where (u) and (v) are either constant integers or single - variable expressions.

1. what are (u) and (v)?

choose 1 answer:
a (u = 3x) and (v = y^4)
b (u = 3x) and (v = y^8)
c (u = 3x^2) and (v = y^4)
d (u = 3x^2) and (v = y^8)
e (u = 9x^4) and (v = y^4)
f (u = 9x^4) and (v = y^8)

1. factor the expression.

Explanation:

1)

To factor an expression as \((U - V)^2\), we use the perfect square trinomial formula \((a - b)^2=a^2 - 2ab + b^2\). We assume the original expression (not fully shown but implied by the factoring form) is a perfect square trinomial. Let's check each option:

• For option C: If \(U = 3x^2\) and \(V=y^4\), then \((U - V)^2=(3x^2)^2-2(3x^2)(y^4)+(y^4)^2 = 9x^4-6x^2y^4 + y^8\) (assuming the original expression is of this form). Let's verify the squares: \((3x^2)^2 = 9x^4\) and \((y^4)^2=y^8\), and the middle term is \(- 2\times(3x^2)\times(y^4)=-6x^2y^4\). This fits the perfect - square trinomial form.
• For other options:
• Option A: \((3x - y^4)^2=9x^2-6xy^4 + y^8\), the first term is \(9x^2\) not matching the expected \(9x^4\) (if the original expression has \(x\) with degree 4).
• Option B: \((3x - y^8)^2=9x^2-6xy^8 + y^{16}\), first term \(9x^2\) and last term \(y^{16}\) do not match.
• Option D: \((3x^2 - y^8)^2 = 9x^4-6x^2y^8 + y^{16}\), last term \(y^{16}\) does not match.
• Option E: \((9x^4 - y^4)^2=81x^8-18x^4y^4 + y^8\), first term \(81x^8\) does not match.
• Option F: \((9x^4 - y^8)^2=81x^8-18x^4y^8 + y^{16}\), first and last terms do not match.

Step 1: Recall the perfect - square formula

The formula for \((a - b)^2\) is \(a^2-2ab + b^2\). From part 1, we know that \(U = 3x^2\) and \(V = y^4\).

Step 2: Apply the formula

Using the formula \((U - V)^2\) with \(U = 3x^2\) and \(V = y^4\), we get \((3x^2-y^4)^2=(3x^2)^2-2\times(3x^2)\times(y^4)+(y^4)^2=9x^4 - 6x^2y^4+y^8\) (assuming the original expression is \(9x^4 - 6x^2y^4 + y^8\)). If we just need to factor the expression as \((U - V)^2\) with \(U = 3x^2\) and \(V = y^4\), the factored form is \((3x^2 - y^4)^2\).

Answer:

C. \(U = 3x^2\) and \(V = y^4\)

2)