1. ## [SOLVED] Repeated factor?

There's this question that baffles me.

Determine whether 2x³ - 3x² + 1 has a repeated factor. If so, find the repeated factor.

I used P(1) and found out that 2x³ - 3x² + 1 = 0 when I substitute the 1 into x. Hence, I know that x - 1 is a factor of 2x³ - 3x² + 1. But how do I know whether x - 1 is a repeated factor or not?

2. $\displaystyle f(x)=2x^3-3x^2+1$

ok x-1 is a factor
so by long divided I do not know what you call it you have

$\displaystyle (x-1)(2x^2-x-1)=2x^3-3x^2+1$

the right side is (2x+1)(x-1)so

$\displaystyle (x-1)(x-1)(2x+1)$

so you have repeated factor

3. Oh, thank you! So I have factorise the quotient and see if one of the factor matches or not? But if I found 2 other factors since this is a cubic equation, does that mean I have to do 3 long divisions in order to see which one is a repeated factor? Or...are there any other less time-consuming way to find out the repeated factor? Thanks!

4. You don't need to do long division. Just do synthetic division. For some examples as to how synthetic division works you could look at this thread:
http://www.mathhelpforum.com/math-he...step-step.html

01

5. Originally Posted by mark1950
Oh, thank you! So I have factorise the quotient and see if one of the factor matches or not? But if I found 2 other factors since this is a cubic equation, does that mean I have to do 3 long divisions in order to see which one is a repeated factor? Or...are there any other less time-consuming way to find out the repeated factor? Thanks!
There is one easy method, if you know differentiation. A root of a polynomial is a repeated root if it is also a zero of the differentiated polynomial.

6. There is one easy method, if you know differentiation. A root of a polynomial is a repeated root if it is also a zero of the differentiated polynomial.
Differentiating polynomials? Yea, I know differentiation and am wondering if it is much different that what I know. If you don't mind, can you please perform the example using this question:

Determine whether 2x³ - 3x² + 1 has a repeated factor. If so, find the repeated factor.

Thank you very much!

7. Originally Posted by mark1950

Determine whether 2x³ - 3x² + 1 has a repeated factor. If so, find the repeated factor.

Thank you very much!
$\displaystyle f(x)=2x^3-3x^2+1$

$\displaystyle f(1)=0$

$\displaystyle f'(x)=6x^2-6x$

$\displaystyle f'(1)=0$

So 1 is a repeated root.

$\displaystyle 2x^3-3x^2+1=(x-1)(x-1)(2x+1)$

8. Thanks. You guys are really much of help!