Question

How do you factor `4m^4 - 37m^2 + 9`?

Answer 1

`(4m^2 - 1)(m^2 - 9)`

Explanation:

Factoring this polynomial gives:

`(4m^2 - 1)(m^2 - 9)`

Answer 2

`4m^4-37m^2+9 = (2m-1)(2m+1)(m-3)(m+3)`

Explanation:

The difference of squares identity can be written:

`a^2-b^2 = (a-b)(a+b)`

We will use this a couple of times, but first note that:

`4m^4-37m^2+9 = 4(m^2)^2-37(m^2)+9`

So we can treat this quartic as a quadratic in `m^2`

Factor using an AC method:

Look for a pair of factors of `AC = 4*9 = 36` with sum `B=37`

The pair `36, 1` works.

Use this pair to split the middle term and factor by grouping:

`4(m^2)^2-37(m^2)+9 = 4(m^2)^2-36(m^2)-(m^2)+9`

`color(white)(4(m^2)^2-37(m^2)+9) = (4(m^2)^2-36(m^2))-((m^2)-9)`

`color(white)(4(m^2)^2-37(m^2)+9) = 4m^2(m^2-9)-1(m^2-9)`

`color(white)(4(m^2)^2-37(m^2)+9) = (4m^2-1)(m^2-9)`

`color(white)(4(m^2)^2-37(m^2)+9) = ((2m)^2-1^2)(m^2-3^2)`

`color(white)(4(m^2)^2-37(m^2)+9) = (2m-1)(2m+1)(m-3)(m+3)`