# [SOLVED] Set of values?

• Jun 7th 2009, 07:17 AM
mark1950
[SOLVED] Set of values?
How to find the set of values of P(x) = 3x³ + 29x² + 65x - 25 given that (x+5) is a repeated factor and (3x - 1) is the third factor and P(x) > 0? It seems like easy because the original question was about finding the factors and as you can see, I've found them so you guys don't need to waste time finding them. The thing I'm confused about now is how to get the set values? My answer was -5 < x < 1/3 but the answer they gave is x > 1/3.

P.s., if my answer is in the wrong section, please notify me as soon as possible but I'm sure its in the correct section because it does not involves differentiation and I've searched and found that many questions of set values are under pre-calculus.
• Jun 7th 2009, 07:55 AM
running-gag
Hi

Is your question to find the set of values of P(x) = 3x³ + 29x² + 65x - 25 > 0 ?

As you said P(x) = (3x-1)(x+5)² and (x+5)² > 0
Therefore P(x) > 0 for 3x-1 > 0 or x > 1/3
• Jun 7th 2009, 07:58 AM
Soroban
Hello, Mark!

Quote:

Find the set of values so that $\displaystyle P(x) \:=\: 3x^3 + 29x^2 + 65x - 25 \:>\:0$
given that $\displaystyle (x+5)$ is a repeated factor and $\displaystyle (3x - 1)$ is the third factor.

They factored the polynomial for us: .$\displaystyle P(x) \;=\;(x+5)^2(3x-1)$

When is the polynomial greater than zero?

It is equal to zero when: .$\displaystyle (x+5)^2(3x-1) \:=\:0$
. . which has roots: .$\displaystyle x \:=\:\text{-}5,\:\frac{1}{3}$

These two values divide the number line into three intervals:

. . $\displaystyle \underbrace{\; - - -\; }\;(\text{-}5)\;\underbrace{\; - - -\;}\;\left(\tfrac{1}{3}\right)\;\underbrace{\; - - -\; }$

Test a value in each interval:

. . $\displaystyle \begin{array}{ccccc}P(\text{-}6) &=& (\text{-}1)^2(\text{-}19) &=& \text{neg} \\ \\[-4mm] P(0) &=& (5)^2(\text{-}1) &=& \text{neg} \\ \\[-4mm] P(1) &=& (6^2)(1) &=& \text{pos} \end{array}$

Therefore, $\displaystyle P(x)$ is positive for: .$\displaystyle x \:>\:\tfrac{1}{3}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If we graph the polynomial equation, the answer is obvious . . .
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• Jun 7th 2009, 08:34 AM
mark1950
Oh! Thanks. So, when two factors are repeated, they meet at the same point on the x-axis...I see...thanks again! Wow...you even showed how to test it...I've never thought of that lol. Thank you once again very much!