If we divide 24 by 6, we get 4, which is the other factor of 24 when given the one factor of 6. Same goes here. In order to find out what the other factor of $y^2-10y+24=0$ is when given one factor of y - 4, we simply divide the second degree polynomial by y - 4 to get the quotient. The quotient, then, is the other factor. Synthetic division is the easiest way to do this.

If y - 4 is the factor, then by the Zero Product Property, y - 4 = 0 and y = 4. Setting up synthetic division:

4| 1 -10 24

____________

The rule is to start by bringing down the first term, multiplying it by the number outside, then putting that product up under the next term in line:

4| 1 -10 24

Then add the column, multiply the sum by the number outside, and put that product up under the next term in line:

4| 1 -10 24

1 -6

And add the last column and that is the remainder. We get a 0 remainder. That means that y - 4 goes evenly into the polynomial and the other factor we are looking for is found in the numbers under the addition line. These numbers are the leading coefficients of the depressed polynomial, the polynomial that serves as the other factor: 1y - 6.

Therefore, the 2 factors that multiply together to give us $y^2-10y+24=0$ are (y - 4)(y - 6) and we can check ourselves by multiplying this out by FOILing to see if the result is the polynomial we started with. It is, so we're all done!

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3 years ago