Synthetic Division: Calulations with Square Roots
Purplemath
Why do square roots show up in synthetic division?
When using synthetic division to factor a polynomial, you will sometimes be given an initial root that contains a square root. These roots do not "just show up"; instead, the author of the exercise constructed a quadratic factor for the polynomial which itself had messy roots.
In other words, and in practical terms, you'll never come up with a square-root test zero; the Rational Roots Test only provides fractional and whole-number test zeroes. You'll only ever work with radical zeroes if they give you one and tell you to start with that. You'll only ever (on your own) find zeroes with square roots after you've found all the regular roots, and find yourself left with that quadratic factor that the author had used for his messy factors.
What do you do, after dividing out the radical factor?
When you have been provided with a radical expression for one of the zeroes of a polynomial, and after you've divided out that factor, your next step will be to divide out the conjugate. That is, when they've given you as one zero, then your next step will be to divide out . (Recall that the conjugate is the exact same two terms, but the sign in the middle has been flipped.)
- Given that
is a zero of x4 + 6x3 − 7x2 − 30x + 10, fully solve the equation x4 + 6x3 − 7x2 − 30x + 10 = 0.
I know that I need to be very careful working problems like this. It's easy to make mistakes.
I do the first couple of steps:
On scratch-paper, I multiply to get the entry that goes under the −7:
Then I do the next couple of steps:
Again, I pull out the scratch-paper and multiply to get the entry that goes under the 10:
Now I can complete this first division:
Since the first zero was , then the next zero I should do is the conjugate, .
With this knowledge, I'm ready for the next division, which works out like this:
Note that, since this second synthetic division handled the conjugate radical root, all the complicated coefficients with radicals disappeared. You should expect this to happen. Whenever you have roots that are conjugates, dividing out one of those roots will make things very messy, but then dividing out the other radical root will clean things back up.
If you want to get the right answers, do not try to do the messier parts in your head or in the margins; take out a sheet of scratch paper and do your work properly.
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