All Questions
68
questions
5
votes
1
answer
242
views
Is there an easy way to tell whether a cubic or quartic polynomial is factorable over the integers?
For a quadratic, it is easy to tell whether it is factorable. If the discriminant is a perfect square, the quadratic is factorable. Otherwise, the quadratic is not factorable. Is there anything ...
- 81
0
votes
2
answers
192
views
Factoring $x^3-6x^2+11x-6$ without using Rational Roots Theorem
To factor $P(x)=x^3-6x^2+11x-6$ one way is using Rational Roots Theorem and recognizing that $x=1$ makes $P(x)$ zero. But I want to factor without using it. I tried,
$$x^3-6x^2+11x-6=x^2(x-6)+11(x-6)+...
- 1,159
2
votes
0
answers
64
views
Irreducibility testing for integral cubic polynomials
I am interested in an algorithm (or irreducibility criterion) for testing whether a cubic polynomial $f \in \mathbb{Q}[x]$ is reducible or not. By Gauss's Lemma, we can work with cubics in $\mathbb{Z}[...
- 325
0
votes
1
answer
68
views
Factoring into a product with three term
I want to solve the following equation:
$$144s^3-408s^2+349s-85 = 0$$
I know that the solution is:
$$(s-1)(12s-17)(12s-5)=0$$
which implies gives, $s=1$ or $s=\dfrac{17}{12} $ or $s=\dfrac{5}{12}$.
...
- 1
-1
votes
1
answer
64
views
If $a\neq 2$ and $a^3+a^2-a-10 = 0$, then evaluate $a+\frac{5}{a}$ [closed]
If $a\neq 2$ and $a^{3}+a^{2}-a-10 = 0$, then what is the value of $a+\dfrac{5}{a}$?
I have plugged equation on WolframAlpha and get the complex values of $a$ but you supposed to do this problem ...
- 141
0
votes
1
answer
423
views
Factor simple cubic polynomial $ax^3 + cx + d = 0$
I'm working on a self-made problem where I have a cubic equation I need to factor. I have some flexibility in choosing my own coefficients for the polynomial, so I am strategically trying to choose my ...
- 416
-1
votes
1
answer
54
views
Factorizing $12x^3 - 8x^2 - 3x + 2$ [closed]
How should I factorize this cubic expression?
$$12x^3 - 8x^2 - 3x + 2$$
- 49
0
votes
3
answers
174
views
Foolproof method for simplifying polynomials with four terms?
When simplifying quadratic equations you have two options:
factoring (which may or may not work)
or the quadratic formula (which will always find the answer)
For quadrinomials what is the go to ...
- 295
5
votes
7
answers
260
views
Solve $\frac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=0$
Solve $$\dfrac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=0.$$
We have $D_x:\begin{cases}x^2-5x+4\ge0\\x^2-5x+4\ne0\end{cases}\iff x^2-5x+4>0\iff x\in(-\infty;1)\cup(4;+\infty).$ Now I am trying to solve the ...
- 5,352
1
vote
2
answers
137
views
Find the zeros of $f(x)=x^3−4x^2+x−4$
I am to find the zeros and multiplicities of $f(x)=x^3−4x^2+x−4$.
The solution provided in the answers section of my book is 4 with multiplicity 1. I arrived at $2\pm\sqrt(8)$.
My working:
$$x^3-4x^2+...
- 2,266
0
votes
1
answer
82
views
about the complex number
Let $a+ ib$ be the complex root of $f(x)=x^3+2x+1$.
I want to find $a$.
My Try: $f(a+ib)=0$. It follow that
$$(a^3-3ab^2+2a+1)=(-b^3+3a^2b+2b)=0$$
- 505
1
vote
3
answers
89
views
A series of multiplication leads to $\frac{1}{2} = 2$
I'm presented with the equation $\frac{a+b}{a} = \frac{b}{a+b}$
Performing cross multiplication yields $a^2+2ab+b^2 = ab$
Subtracting $ab$ from both sides, we get $a^2+ab+b^2 = 0$
Multiplying both ...
- 160
0
votes
2
answers
348
views
Calculating eigenvalues for a $3 \times 3$ matrix without solving a cubic
I am trying to find the eigenvalue for question (h), however I am unable to factor out and find the eigenvalues(roots) after I take the determinant of the characteristic equation.
Let $x$ be an ...
- 11
2
votes
2
answers
148
views
Solving degree 3 equations
Solve for $x$,
\begin{cases}4x^3+3x^2y+y^3=8\\
2x^3-2x^2y+xy^2=1\end{cases}
I tried substitution of $x$, but it got very complex.
Is there a simpler way to do this?
user751264
1
vote
3
answers
374
views
Finding a cubic formula for roots of cubic equations
Solve for $x$,
$$27x^3+21x+8=0$$
I would like to know if there exists an formula for cubic equations just like quadratic formula for quadratic equations.
user751264