how do you factor 16x^4 - 40x^2 + 9 I don't understand how the factoring should go. When I think of a way to factor it it doesn't turn out right.
(4x^2 - 9)(2x^2 - 1)
= (2x + 3)(2x - 3)(2x^2 - 1)
16x^4 - 40x^2 + 9 = (4x^2 -9)(4x^2 -1)
= (2x+3)(2x-3)(2x+1)(2x-1)
16x^2-121
To factor the quadratic expression 16x^4 - 40x^2 + 9, we can follow a method called factoring by grouping. Here's how you can do it step by step:
Step 1: Identify any common factors.
Check if there's a common factor among all the terms. In this case, there are no common factors among the terms.
Step 2: Split the middle term.
To factor the quadratic, we'll split the middle term (-40x^2) into two terms whose coefficients multiply to give the product of the first and last term. The first term is 16x^4, and the last term is 9.
The product of 16x^4 and 9 is 144x^4. We need to find two numbers that have a product of 144 and add up to -40 (the coefficient of the middle term). After a little trial and error, we find that -36 and -4 satisfy this requirement.
So we split the middle term -40x^2 into -36x^2 - 4x^2:
16x^4 - 36x^2 - 4x^2 + 9
Step 3: Group the terms.
Group the first two terms and the last two terms:
(16x^4 - 36x^2) + (-4x^2 + 9)
Step 4: Factor out common terms from each group.
From the first group (16x^4 - 36x^2), we can factor out the greatest common factor, which is 4x^2:
4x^2(4x^2 - 9)
From the second group (-4x^2 + 9), notice that it is in the form of difference of squares. We can factor it using that pattern:
-(2x - 3)(2x + 3)
After factoring out the common terms, we have:
4x^2(4x^2 - 9) - (2x - 3)(2x + 3)
Step 5: Factor further if possible.
Now, we can notice that both terms have a common factor of (4x^2 - 9). It is a difference of squares, which can be factored using the formula: a^2 - b^2 = (a + b)(a - b).
So, we have:
4x^2(2x - 3)(2x + 3) - (2x - 3)(2x + 3)
Step 6: Apply the distributive property.
Finally, apply the distributive property to simplify further:
(4x^2 - 1)(2x - 3)(2x + 3)
Therefore, the factored form of 16x^4 - 40x^2 + 9 is (4x^2 - 1)(2x - 3)(2x + 3).