Question

Quentin solved the polynomial equation
x
4
−
12
x
2
+
56
=
20
x
2
−
200
using the steps shown below. We know for sure that Step 1 is correct.

x
4
−
32
x
2
+
56
=
−
200
x
4
−
32
x
2
+
256
=
0
(
x
2
−
16
)
2
=
0
(
(
x
+
4
)
(
x
−
4
)
)
2
=
0
(
x
+
4
)
(
x
−
4
)
(
x
+
4
)
(
x
−
4
)
=
0
x
=
−
4
,
4
Select three statements that correctly justify the steps of Quentin's solution process.


Step 2 applies the Addition Property of Equality.


Step 3 applies the difference of squares formula to factor the expression.


Step 4 applies the difference of squares formula to factor the expression inside the parentheses.


Step 5 applies the Distributive Property.


Step 6 applies the Remainder Theorem to find the solutions of the equation in factored form.

Answer 1

Only two of the given statements are correct.

Correct:
- Step 2 applies the Addition Property of Equality. (200 was added to both sides to get x^4 − 32x^2 + 256 = 0.)
- Step 4 applies the difference of squares formula to factor the expression inside the parentheses. (x^2 − 16 = (x + 4)(x − 4).)

Why the others are incorrect:
- Step 3 does not use the difference of squares; it recognizes a perfect-square trinomial: x^4 − 32x^2 + 256 = (x^2 − 16)^2.
- Step 5 is not an application of the Distributive Property; it uses the exponent/property (A)^2 = A·A to write ((x+4)(x−4))^2 as (x+4)(x−4)(x+4)(x−4).
- Step 6 does not use the Remainder Theorem; the solutions follow from the Zero Product Property applied to the factored form.