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
Pattern #3: (a−b)(a+b)=a2−b2
In this case, the pattern that can be used to factor the high-degree polynomial is Pattern #2: (a−b)2=a2−2ab+b2
To factor the high-degree polynomial y^4 - 8y^2 + 16, we can use Pattern #3:
(a - b)(a + b) = a^2 - b^2
In this case, let's identify y^2 as 'a' and 4 as 'b'.
Therefore, we have:
(a - b)(a + b) = (y^2 - 4)(y^2 + 4)
= (y^2 - 2^2)(y^2 + 2^2)
= (y^2 - 2)(y^2 + 2)(y^2 + 2)
The factors of the high-degree polynomial are (y^2 - 2)(y^2 + 2)(y^2 + 2).
To factor the high-degree polynomial y^4 - 8y^2 + 16, we should look for a pattern that matches the form of the polynomial.
One pattern that can be used is Pattern #3: (a - b)(a + b) = a^2 - b^2.
To apply this pattern to the given polynomial, we can rewrite it as (y^2)^2 - 4^2. By matching the form of the pattern, we can see that a = y^2 and b = 4.
Therefore, using the pattern, we can factor the polynomial as follows:
y^4 - 8y^2 + 16 = (y^2 - 4)(y^2 + 4).
Note that the factors (y^2 - 4) and (y^2 + 4) cannot be further factored over the real numbers.