Question

Find the second factor of 9x^2 + 16 + 1

Answer 1

`\sf (3x + 1)`

Explanation:

To find the second factor, we need to determine what to multiply
`\sf (3x + 1)` by to get the given product
`\sf 9x^2 + 6x + 1`.

Let the second factor be
`\sf (ax + b)`. Now, we can perform the multiplication and equate it to the given product:

`\sf (3x + 1)(ax + b) = 9x^2 + 6x + 1`

Now, distribute the terms:

`\sf 3x(ax + b) + 1(ax + b) = 9x^2 + 6x + 1`

Simplify further:

`\sf 3ax^2 + 3bx + ax + b = 9x^2 + 6x + 1`

Now, equate the corresponding coefficients:

Coefficient of
`\sf x^2`:
`\sf 3a = 9`

`\sf \Rightarrow`
`\sf a =(9)/(3)= 3`

2. Coefficient of
`\sf x`:
`\sf 3b + a = 6`

`\sf \Rightarrow`
`\sf 3b + 3 = 6`

`\sf \Rightarrow`
`\sf b =(6-3)/(3)=1`

So, the second factor is
`\sf (ax + b) = (3x + 1)`.