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laurakate
01-31-2010, 02:46 PM
Factor the polynomial 3x^3 + 2x^2 - 7x + 2 completely using the following steps.
1) Explain the connection between the roots of a polynomial and its factors
2) Find a root of the given polynomial by guessing or by trial-and-error
3) Using the root you found above give a factor of the polynomial
4) Divide the polynomial by the factor you found to demonstrate that it is a genuine factor and obtain its remaining factor
5) Factor the remaining factor you got above completely and give a complete factorization of the whole polynomial.

mmm4444bot
01-31-2010, 03:32 PM
3x^3 + 2x^2 - 7x + 2

Find a root of the given polynomial by guessing or by trial-and-error

I think guessing IS trial-and-error.

Did you try this, yet?

Here is some basic information that you should learn.

If some number r is a root of a polynomial, then the expression x - r is a factor of the polynomial.

Once you know one factor of a polynomial, you can use division to find remaining factors.

EG:

x^3 - 37x - 84

r = -3 is a root of this polynomial because substituting -3 for x causes the polynomial to evaluate to zero. (That's the definition of a root.)

(-3)^3 - 37(-3) - 84 = 0

Well, if -3 is a root, then x - (-3) is a factor of the polynomial. In other words, the expression x + 3 evenly divides the polynomial.

Using polynomial long-division, we get the following.

(x^3 - 37x - 84)/(x + 3) = x^2 - 3x - 28

The two factors of this quadratic quotient can be found using your favorite method.

Again, please ask specific questions, so that we might determine where to begin helping you.

Cheers ~ Mark

laurakate
02-01-2010, 10:22 AM
Sorry about not putting any questions. I don't want anyone to do my homework for me! I think I have the answers to 1 and 2, but 3, 4, and 5 are confusing me. I got x=1 as a factor and x-1 as a root. ( I think this is correct). I divided and got a remainder of 0. Am I on the right track? What I am I supposed to do for5 ? Isn't this the same step I did in d?

Thanks for any help or guidance and I apologize again for not posting what exactly I needed help on!

mmm4444bot
02-01-2010, 02:39 PM
I got x=1 as a factor and x-1 as a root. The number 1 is correct, but your terminology is backwards.

x = 1 is a root of the polynomial 3x^3 + 2x^2 - 7x + 2 (Good job.)

x - 1 is a factor of the polynomial 3x^3 + 2x^2 - 7x + 2

You still need help with (3) ?

3) Using the root you found above give a factor of the polynomial You already answered this question (see above).

4) Divide the polynomial by the factor you found to demonstrate that it is a genuine factor and obtain its remaining factor

You said that you got a remainder of zero, when you did this division. This is sufficient to show that the divisor is a factor of the dividend. If x-1 were not a factor, then it would not divide evenly.

5) Factor the remaining factor you got above completely and give a complete factorization of the whole polynomial.

If you did the division properly, the quotient is a quadratic polynomial. Can you factor it? You will then have all three factors of the original polynomial.

What I am I supposed to do for5 ? Isn't this the same step I did in d? What is d?

laurakate
02-01-2010, 09:01 PM
Thank you so much. You helped me to answer this entire problem with confidence! I get confused with the wording of these problems and I feel very intimidated. Also, I didn't understand the connection between roots and polynomials. Now that you showed me the examples and explained it to me...I get it! Thanks!

Thank you again!