# Factor: 4x^2 - 16t^2  ; -10y^2 + 31y - 15

embizze | High School Teacher | (Level 1) Educator Emeritus

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(1) `4x^2-16t^2` is an example of the difference of two squares.

`a^2-b^2=(a+b)(a-b)`  Here a=2x and b=4t so:

`4x^2-16t^2=(2x+4t)(2x-4t)` Factoring out the common 2 we get:

`=2(x+2t)2(x-2t)`

`=4(x+2t)(x-2t)`

So the factorization is 4(x+2t)(x-2t)

Note that we could simplify the process by factoring out the common 4 first:

`4x^2-16t^2=4(x^2-4t^2)=4(x+2t)(x-2t)`

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(2) `-10y^2+31y-15`

- there are no common factors of the terms

- this is not a special product pattern

One method is the following:

given `ax^2+bx+c`

(a) Find the product `ac`

(b) Find two numbers `p,q` such that `pq=ac` and `p+q=b`

(c) rewrite as `ax^2+px+qx+c`

(d) factor in pairs

Here `-10y^2+31y-15=-[10y^2-31y+15]`

`10*15=150` and `-25*-6=150,-25+-6=-31` so:

`-[10y^2-31y+15]=-[10y^2-6y-25y+15]`

`=-[2y(5y-3)-5(5y-3)]`

`=-[(5y-3)(2y-5)]`

So the factorization is -(5y-3)(2y-5)