Question

How do you factor `a^2b+6ab^2+9b^2`?

Answer 1

See solution below

Explanation:

To factor this, because all terms are positive and there are two variables we know the factoring will be in the form:

`(color(red)(j)a + color(blue)(k)b)(color(green)(l)a + color(purple)(m)b)`

Where `color(red)(j)`, `color(blue)(k)`, `color(green)(l)` and `color(purple)(m)` are constants.

We also know because of the coefficients in the original problem:
`color(red)(j) xx color(green)(l) = 1`
`color(blue)(k) xx color(purple)(m) = 9`
`(color(red)(j) xx color(purple)(m)) + (color(blue)(k) xx color(green)(l)) = 6`

"Playing" with the factors for `9` (1x9, 3x3, 9x1) Gives:

`(color(red)(1)a + color(blue)(3)b)(color(green)(1)a + color(purple)(3)b)`

`(a + 3b)(a + 3b) = (a + 3b)^2`