Identify the pattern that can be used to factor the high-degree polynomial y4−8y2+16
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Pattern #1: (a+b)2=a2+2ab+b2
Pattern #2: (a−b)2=a2−2ab+b2
The pattern that can be used to factor the high-degree polynomial y4−8y2+16 is Pattern #1: (a+b)2=a2+2ab+b2.
To factor the high-degree polynomial y^4 - 8y^2 + 16, we can utilize Pattern #2, which states that (a-b)^2 = a^2 - 2ab + b^2.
In this case, let's rewrite the polynomial as (y^2)^2 - 2*(y^2)*4 + 4^2.
Using Pattern #2, we can rewrite it as (y^2 - 4)^2.
Therefore, the factored form of the polynomial y^4 - 8y^2 + 16 is (y^2 - 4)^2.
To factor the high-degree polynomial y^4 - 8y^2 + 16, we can use pattern #2: (a - b)^2 = a^2 - 2ab + b^2.
Let's break down the polynomial and see if it fits the pattern:
a = y^2
b = 4
Using the pattern, we can substitute these values into the formula:
(a - b)^2 = a^2 - 2ab + b^2
(y^2 - 4)^2 = (y^2)^2 - 2(y^2)(4) + (4)^2
(y^2 - 4)^2 = y^4 - 8y^2 + 16
As we can see, the polynomial y^4 - 8y^2 + 16 satisfies the pattern (a - b)^2 = a^2 - 2ab + b^2 when we choose a = y^2 and b = 4. Therefore, we can factor the polynomial as (y^2 - 4)^2.