To find the completely factored form of the polynomial x4 – 8x2 + 9, we can use substitution to simplify our work.
Let’s set y = x2. Then the expression becomes:
y2 – 8y + 9
Next, we want to factor this quadratic expression. We need to find two numbers that multiply to 9 (the constant term) and add up to -8 (the coefficient of the middle term). The numbers that meet these criteria are -3 and -3.
This allows us to factor the quadratic as follows:
(y – 3)(y – 3) = (y – 3)2
Substituting back for y, we have:
(x2 – 3)2
Thus, the completely factored form of the original polynomial is:
(x2 – 3)2