Question

9x⁴– 6x²y4 + y8

We can factor the expression as (U – V)2 where U and V are either constant integers or single-variable expressions. What are U and V?

Answer 1

To factor the quartic polynomial 9x⁴ − 6x²y⁴ + y⁸ as (U − V)², U is (3x² + y⁴) and V is 3x²y⁴.

Step-by-step explanation:

The expression given is a quartic polynomial in the form of 9x⁴ − 6x²y⁴ + y⁸ and can be factored into a perfect square difference of two squares. To factor the expression as (U − V)², we identify U and V as two binomials such that their square difference yields the original expression.

Let's start by recognizing that each term in the expression is a perfect square:

Thus, we can rewrite the given expression as:

(3x² + y⁴)² - (2 · 3x² · y⁴)

Comparing this to the difference of squares (U − V)², we can identify:

Therefore, the factored form of the expression is (3x² + y⁴ - 3x²y⁴)², where U is (3x² + y⁴) and V is 3x²y⁴.

Answer 2

Answer:The given expression is

9

�

4

−

6

�

2

�

4

+

�

8

9x

4

−6x

2

y

4

+y

8

.

To factor the expression as

(

�

−

�

)

2

(U−V)

2

, we need to identify two terms whose squares create the given expression.

Let's try to express the given expression as the square of a binomial:

(

�

−

�

)

2

=

�

2

−

2

�

�

+

�

2

(U−V)

2

=U

2

−2UV+V

2

Now, let's compare this with the given expression:

�

2

−

2

�

�

+

�

2

=

9

�

4

−

6

�

2

�

4

+

�

8

U

2

−2UV+V

2

=9x

4

−6x

2

y

4

+y

8

Comparing coefficients, we can identify

�

U and

�

V:

�

2

U

2

corresponds to

9

�

4

9x

4

, so

�

=

3

�

2

U=3x

2

.

�

2

V

2

corresponds to

�

8

y

8

, so

�

=

�

4

V=y

4

.

Now, let's substitute these values back into the binomial expression:

(

�

−

�

)

2

=

(

3

�

2

−

�

4

)

2

(U−V)

2

=(3x

2

−y

4

)

2

Therefore,

�

=

3

�

2

U=3x

2

and

�

=

�

4

V=y

4

are the values that make the given expression equal to

(

�

−

�

)

2

(U−V)

2

.

Step-by-step explanation: