Question

The polynomial 6x2 + x − 15 has a factor of 2x − 3. What is the other factor?

Answer 1

Explanation:

If you want to know what you have to multiply by 5 to get a product of 70, you would divide 70 by 5 to get that other number. Because 70 divided by 5 is 14, you know that the other number is 14. It's the same when you have one factor of a polynomial and want to know the other. You divide your polynomial by the one factor to get to the other factor, because when you multiply them together, you get back the polynomial you started with. Like when we found that 70 / 5 = 14, we know that when we multiply 14 by 5 we'll get the 70 we started with.

Long division of polynomials will be difficult to illustrate within this forum, but I'll do my best. What we want to do is divide
`6x^2+x-15` by 2x - 3:

__________

2x - 3 |
`6x^2+x-15`

First divide the 6 x-squared by the 2x (forget about the rest of what's inside the box for now, and also forget about the -3 for now. We'll deal with them later). 2x goes into 6 x-squared 3x times, because 2x * 3x = 6x-squared, right? So the 3x goes above the 6x-squared:

3x_______

2x - 3 |
`6x^2+x-15`

Now we multiply the 3x by the 2x and put that product under the 6x-squared. The product is 6x-squared (which will always be the case in division that they will subtract each other away. That's why we do this!)Then multiply the 3x by the -3 which is -9 and put that under the "+x" term:

3x________

2x - 3 |
`6x^2+x-15`

6x^2-9x

At this point we change the signs and have

3x

2x - 3 | 6x^2 + x - 15

- 6x^2 +9x

And then we add and bring down the -15:

3x

2x - 3 | 6x^2 + x - 15

- 6x^2 + 9x

10x - 15

Now do the division process again, this time dividing 10x by 2x (again the -3 will wait). 10x divided by 2x is 5 (because 2x * 5 = 10x, right?). The 5 goes above the +x on top:

3x + 5

2x - 3 | 6x^2 + x - 15

- 6x^2 + 9x

10x - 15

Now we wwill multiply the 5 by both the 2x and the -3 and subtract:

3x + 5

2x - 3 | 6x^2 + x - 15

- 6x^2 + 9x

10x - 15

- 10x + 15

Notice that when we subtracted, the 10x changed from + to -, and the -15 changed to a positive. The remainder is 0.

That tells us that the other factor of the polynomial is 3x + 5. You could also have done synthetic division, but it would be more difficult than this to illustrate because you would be dividing by 3/2. :/