Factoring Polynomial: Get Expert Help Now!

[bergausstein]

Please assist me in this problem.$\frac{2}{3}b^5-\frac{1}{6}b^3+\frac{4}{9}b^2-1$

I tried grouping but still could not find anything factorable form of the expression.

Regards.

 

[Jameson]

Please assist me in this problem.$\frac{2}{3}b^5-\frac{1}{6}b^3+\frac{4}{9}b^2-1$

I tried grouping but still could not find anything factorable form of the expression.

Regards.

Hi bergausstein,

After thinking it over and cheating a bit (using Wolfram), I'm not sure if a great factorization exists for this polynomial. You can definitely rewrite it a bit, but simpler is relative here. Is this problem from a textbook? What methods are you covering? I'm curious how simplified the answer is intended to be.

 

[Fernando Revilla]

As Jameson says, a great factorization is impossible for this polynomial. We can write $$p(x)=(2/3)x^5-(1/6)x^3+(4/9)x^2-1=\frac{1}{18}(\underbrace{12x^5-3x^3+8x^2-18}_{q(x)}).$$ We have $q(1)=\ldots <0$ and $q(2)=\ldots>0,$ so by Bolzano's theorem there is a root $\xi\in (1,2).$
It is not trivial to prove that $q(x)$ has only one real root, but even in this case and using Ruffini algorthim:

$$\begin{array}{r|rrrrrr}
& 12 & 0 & -3 & 8 & 0 & -18 \\
\xi & & 12\xi & 12\xi^2 & 12\xi^3-3\xi & 12\xi^4-3\xi^2+8\xi & 12\xi^5-3\xi^3+8\xi^2\\
\hline & 12 & 12\xi & 12\xi^2-3 & 12\xi^3-3\xi+8 & 12\xi^4-3\xi^2+8\xi & \underbrace{12\xi^5-3\xi^3+8\xi^2-18}_{=0} \end{array}$$ So, the factorization of $p(x)$ in $\mathbb{R}[x]$ would be $$p(x)=\dfrac{1}{18}(x-\xi)\left(12x^4+12\xi x^3+(12\xi^2-3)x^2+(12\xi^3-3\xi+8)x+12\xi^4-3\xi^3+8\xi\right).$$ That is, a complete disaster. :)

 

[HallsofIvy]

As Fernando Revilla pointed out, we can write this polynomial as \(\displaystyle \frac{1}{18}\left(12x^5- 3x^3+ 8x^2- 18\right)\) so this becomes a question of factoring \(\displaystyle 12x^5- 3x^3+ 8x^2- 18\).

By the "rational root theorem", if there is a rational root (so that there is a factor with integer coefficients) then it must be of the form $$\frac{m}{n}$$ (and the factor of the form (nx- m))where n is an integer factor of the leading coefficient, 12, and m is an integer factor of the constant term, 18. The integer factors of 12 are 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, and -12 and the integer coefficients of 18 are 1, -1, 2, -2, 3, -3, 6, -6, 9, -9, 18, and -18.

Form all the different fractions with those numerators and denominators (there not as many as there might seem- a lot of cancelling occurs) and put them into the polynomial to see If they work.

 

[vkalyan]

Please assist me in this problem.$\frac{2}{3}b^5-\frac{1}{6}b^3+\frac{4}{9}b^2-1$

I tried grouping but still could not find anything factorable form of the expression.

Regards.

Short answer, it is not factorizble with real coefficients. Of course every polynomial with complex roots is factorizable with complex coeffcients. I don't know how to get those complex roots but looks like that's not what you are looking for. And what you seem to be looking for is not possible

 

[bwpbruce]

$\dfrac{2}{3}b^5 - \dfrac{1}{6}b^3+\dfrac{4}{9}b^2 - 1$

$=\dfrac{12}{18}b^5 - \dfrac{3}{18}b^3 + \dfrac{8}{18}b^2 - \dfrac{18}{18}$

$=\dfrac{1}{18}\left(12b^5 - 3b^3 + 8b^2 - 18\right)$

$=\dfrac{1}{18}\left(3b^3(4b^2-1)+2(4b^2-9)\right)$

$=\dfrac{1}{18}\left(3b^3(4b^2-1)+2\left(4b^2-9 \cdot\ \dfrac{4b^2-1}{4b^2-1}\right)\right)$

$=\dfrac{1}{18}\left(3b^3(4b^2-1)+2\left(\dfrac{4b^2-9}{4b^2-1}\right)(4b^2-1)\right)$
$=\dfrac{1}{18}\left((4b^2 - 1)\left(3b^3 + \dfrac{2(4b^2 - 9)}{4b^2 - 1}\right)\right)$
$=\dfrac{1}{18}\left((2b+1)(2b-1)\left(3b^3 + \dfrac{2(2b + 3)(2b-3)}{(2b+1)(2b-1)}\right)\right)$

 

[Fernando Revilla]

$\dfrac{2}{3}b^5 - \dfrac{1}{6}b^3+\dfrac{4}{9}b^2 - 1$
$=\dfrac{1}{18}\left((2b+1)(2b-1)\left(3b^3 + \dfrac{2(2b + 3)(2b-3)}{(2b+1)(2b-1)}\right)\right)$

According to your solution, $b=\pm\dfrac{1}{2}$ are roots. Have you checked them?

 

[bwpbruce]

According to your solution, $b=\pm\dfrac{1}{2}$ are roots. Have you checked them?

Is there not a distinction between factoring a polynomial EXPRESSION and finding the roots of a polynomial EQUATION? Furthermore, even if we were tasked with finding roots, in accordance with the zero product property, $b=\pm\dfrac{1}{2}$ would be $\textit{possible}$ roots. But it is clear that neither value of b would work as a root. It is clear that we have to guess what the root is. If we assume that b = 1, then for the final binomial we have

$3b^3 + \dfrac{2(2b + 3)(2b-3)}{(2b+1)(2b-1)} = 0$
$3(1)^3 + \dfrac{2(2(1) + 3)(2(1)-3)}{(2(1)+1)(2(1)-1)} = 0$

$3(1)^3 + \dfrac{2(5)(-1)}{(3)(1)} = 0$

$3- \dfrac{10}{3} = 0$

$3- 3.33 = 0$

$-(3.33 - 3) = 0$

$-0.33 \approx 0$

So the value of b is close to 1.

 

[Fernando Revilla]

In these problems, we are supposed to provide a closed expression. For example if $p(x)=x^2-2$, then $1.5^2-2=0.25\approx 0.$ A closed expression is $p(x)=(x-\sqrt{2})(x+\sqrt{2})$, not $p(x)=(x-1.5)(x+1.5).$

 

[bwpbruce]

In these problems, we are supposed to provide a closed expression. For example if $p(x)=x^2-2$, then $1.5^2-2=0.25\approx 0.$ A closed expression is $p(x)=(x-\sqrt{2})(x+\sqrt{2})$, not $p(x)=(x-1.5)(x+1.5).$

Which of my expressions are you implying is something other than closed form?

 

[Fernando Revilla]

Which of my expressions are you implying is something other than closed form?

(1) You write $$\dfrac{2}{3}b^5 - \dfrac{1}{6}b^3+\dfrac{4}{9}b^2 - 1=\dfrac{1}{18}\left((2b+1)(2b-1)\left(3b^3 + \dfrac{2(2b + 3)(2b-3)}{(2b+1)(2b-1)}\right)\right)$$ What happens when you substitute $b=\pm\dfrac{1}{2}$?

(2) The title of your thread Factoring polynomial, usually means to express a polynomial as a product of irreducible polynomial and does not include rational fractions. Otherwise, you should specify the terms of the factorization you are referring to.

 

[bwpbruce]

(1) You write $$\dfrac{2}{3}b^5 - \dfrac{1}{6}b^3+\dfrac{4}{9}b^2 - 1=\dfrac{1}{18}\left((2b+1)(2b-1)\left(3b^3 + \dfrac{2(2b + 3)(2b-3)}{(2b+1)(2b-1)}\right)\right)$$ What happens when you substitute $b=\pm\dfrac{1}{2}$?

(2) The title of your thread Factoring polynomial, usually means to express a polynomial as a product of irreducible polynomial and does not include rational fractions. Otherwise, you should specify the terms of the factorization you are referring to.

You're still asking about $b=\pm\dfrac{1}{2}$?.