Question

ALGEBRA AND GEOMETRY REVIEWFactoring a quadratic with a negative leadFactor completely.- 4y2 - 32y +80Х

Answer 1

Factoring polynomials is a matter of making educated guesses. In some cases the educated guesses come easily in others it is not so easy a task.

For our polynomial, let us first factor out 4. Doing this gets is

`-4(y^2+8y-20)`

And now we take a guess at the factors of the polynomial in the parenthesis.

Now remember that,

`(y+a)(y+b)=y^2+(a+b)y+ab`

Comparing this with our polynomial suggests

`a+b=8`
`ab=-20`

meaning the sum of a and b is 8 and the product is -20. What two numbers could satisfy this condition?

Let us take a guess at a = 10 b = -2. This will give

`-4\textcolor{#FF7968}{\lbrack}(y+10)(y-2)\textcolor{#FF7968}{\rbrack}`

Let us see if this guess works by expanding what is in the square brackets.

`\textcolor{#FF7968}{(y+10)(y-2)}=y(y-2)+10(y-2)=y^2-2y+10y-20`
`=\textcolor{#FF7968}{y^2+8y-20}`

which is exactly the same!

Hence, the factorization of - 4y2 - 32y + 80 is

`-4(y+10)(y-2)\text{.}`