All Questions
Tagged with cubics algebra-precalculus
280
questions
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Quicker and non-trivial methods for solving Cubic Equation
Motivation : There have been many elementary ways like Hit-and-trial method, Polynomial division and others used in teaching how to solve cubic equation. I wanted to find a method that is faster to ...
- 95
1
vote
2
answers
62
views
Prove that $a=0$ if and only if $b=0$ for the cubic $x^3 + ax^2 + bx + c=0$ whose roots all have the same absolute value.
Take three real numbers $a, b$ and $c$ such that the roots of equation $x^3+ax^2+bx+c=0$ have the same absolute value. We need to show that $a=0$ if and only if $b=0$.
I tried taking the roots as $p, ...
- 23
2
votes
3
answers
103
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Prove that $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots
I'm trying to prove that the cubic equation
$a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$
has three real roots. The coefficients are
$a_3 = - 1 - \sigma - \tau - \chi$
$a_2 = -2 (\sigma +...
- 61
2
votes
1
answer
121
views
Three real roots of a cubic
Question: If the equation $z^3-mz^2+lz-k=0$ has three real roots, then necessary condition must be _______
$l=1$
$ l \neq 1$
$ m = 1$
$ m \neq 1$
I know there is a question here on stack about ...
- 497
0
votes
2
answers
109
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find $a$ and $b$ where $x^3 - 4x^2 -3x + 18 = (x+a)(x-b)^2$
I have a problem solving this as when I match the coefficients to the expanded brackets I end up with $2$ unknowns $a \& b$. So cannot substitute any values to find the other. According to the ...
1
vote
1
answer
81
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Why Can't Cubic Equation Have Fractional Solutions When Its Coefficients Are All Integers? [duplicate]
In Leonhard Euler's book, "The Elements of Algebra" he seems to say that if we convert any cubic equation into the form $x^3 + ax^2 + bx + c$, and make sure that $a$, $b$ and $c$ are integer ...
- 1,467
5
votes
2
answers
120
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If the complex roots of $x^3-x-2=0$ are $r\pm si$, and $As^6 +Bs^4 + Cs^2 =26$ for integers $A$, $B$, $C$, find $A+B+C$
The question:
In the cubic $x^3-x-2=0$, there is one real root and two complex roots of the form $r\pm si$, with $r$ and $s$ real. If there exists integers $A,B,$ and $C$ such that $As^6 +Bs^4 + Cs^2 ...
- 159
1
vote
1
answer
203
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How do I find a cubic equation given only one root?
Given the root of a cubic equation $Z = \sqrt[3]{Y + \sqrt{Y^2 - \frac{X^6}{27}}} + \sqrt[3]{Y - \sqrt{Y^2 - \frac{X^6}{27}}} - X$ and the assumption that both $X$ and $Y$ are greater than zero, is ...
- 1,861
3
votes
3
answers
165
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Is there any faster way to factor $x^3-3x+2$?
$$x^3-3x+2$$
$$x^3-3x+x^2+2-x^2$$
$$x^2-3x+2+x^3-x^2$$
$$(x-2)(x-1)+x^2(x-1)$$
$$(x-1)[x^2+x-2]$$
$$(x-1)(x+2)(x-1)$$
Is there a better, faster way to factor this cubic trinomial?
- 1,043
1
vote
2
answers
569
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Find sum of all integral values of $r$ such that all roots of the equation $x^3-(r-1)x^2-11x+4r=0$ are also integers
Find sum of all integral values of $r$ such that all roots of the equation $$x^3-(r-1)x^2-11x+4r=0$$
are also integers.
What I could do was $$r=\frac{x^3+x^2-11x}{x^2-4}=x+1+\frac{4-7x}{x^2-4}$$
Since ...
- 9,599
3
votes
6
answers
406
views
Find all real numbers $a$ for equation $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.
Problem:
Find all real numbers $a$ for which the equation, $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.
This problem is from an algebra round of a local high school math competition that has ...
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4
votes
1
answer
259
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Why doesn't simultaneous equations work to find co-efficients of a cubic that passes through four points?
I'm trying to find the equation of a cubic that passes through three specific points (technically it's four but that point is y-intercept). The equation would look something like this:$f(x)=ax^3+bx^2+...
- 43
1
vote
1
answer
67
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Signs in the Cardano formula
When deriving the Cardano formula from
$x^{3}+px+q=0$ we let $x$ be a sum and compare coefficients. So
$x=u+v$, then we get a system for $u$ and $v$.
We get $(1) -q=u^{3}+v^{3}$ and $(2) u^{3}v^{3}=-(\...
2
votes
2
answers
220
views
Determine whether the roots of a cubic equation have positive real part
Consider the cubic equation
$a x^3 + b x^2 + cx + d =0 $,
where all coefficients depend on three parameters
$$a=a(i, j, k), b=b(i, j, k),\cdots$$
and $a, b, c \in \mathbb{R}$ for all $(i, j, k)$. The ...
5
votes
1
answer
127
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Find the value of: $\sqrt[3]{a+b}+\sqrt[3]{b+c}+\sqrt[3]{a+c}$
Let $a,b,c$ be roots of the cubic
$$x^3-x^2-2x+1=0$$
Then, find the value of:
$$\sqrt[3]{a+b}+\sqrt[3]{b+c}+\sqrt[3]{a+c}$$
My attempt.
I used the substitutions $$a+b=x^3, b+c=y^3, a+c=z^3$$
$$x^3+y^...
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