Facorisation of 3a^3 + 4a - 3
Answers
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✤ Required Answer:
✒ To FinD:
Facorisation of 3a^3 + 4a - 3
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✤ How to Solve?
To solve this question, u should know about cubic equation and it's formula..
A cubic formula for the roots of the general cubic equation (with a ≠ 0)
{\displaystyle ax^{3}+bx^{2}+cx+d=0}ax^3+bx^2+cx+d=0
can be deduced from every variant of Cardano's formula by reduction to a depressed cubic. The variant that is presented here is valid not only for real coefficients, but also for coefficients a, b, c, d belonging to any field of characteristic different of 2 and 3.
The formula being rather complicated, it is worth splitting it in smaller formulas.
Let
{\displaystyle {\begin{aligned}\Delta _{0}&=b^{2}-3ac,\\\Delta _{1}&=2b^{3}-9abc+27a^{2}d,\end{aligned}}}{\displaystyle {\begin{aligned}\Delta _{0}&=b^{2}-3ac,\\\Delta _{1}&=2b^{3}-9abc+27a^{2}d,\end{aligned}}}
and
{\displaystyle C={\sqrt[{3}]{\frac {\Delta _{1}\pm {\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}}{2}}},}{\displaystyle C={\sqrt[{3}]{\frac {\Delta _{1}\pm {\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}}{2}}},}
where the symbols {\displaystyle {\sqrt {{~}^{~}}}}{\displaystyle {\sqrt {{~}^{~}}}} and {\displaystyle {\sqrt[{3}]{{~}^{~}}}}{\displaystyle {\sqrt[{3}]{{~}^{~}}}} denote any square root and any cube root, respectively. The sign "±" before the square root is either "+" or "–"; the choice is almost arbitrary, and changing it amounts to change of square root. However, if one choice leads to C = 0, the other sign must be selected. Then, one of the roots is
{\displaystyle x=-{\frac {1}{3a}}\left(b+C+{\frac {\Delta _{0}}{C}}\right){\text{.}}}{\displaystyle x=-{\frac {1}{3a}}\left(b+C+{\frac {\Delta _{0}}{C}}\right){\text{.}}}
The other two roots can be obtained by changing the choice of the cube root in the definition of C, or, equivalently by multiplying C by a primitive cube root of unity, that is
–1 ± √–3
/
2
. In other words, the three roots are
{\displaystyle x_{k}=-{\frac {1}{3a}}\left(b+\xi ^{k}C+{\frac {\Delta _{0}}{\xi ^{k}C}}\right),\qquad k\in \{0,1,2\}{\text{,}}}{\displaystyle x_{k}=-{\frac {1}{3a}}\left(b+\xi ^{k}C+{\frac {\Delta _{0}}{\xi ^{k}C}}\right),\qquad k\in \{0,1,2\}{\text{,}}}
where ξ =
–1 + √–3
/
2
.
As for the special case of a depressed cubic, this formula applies but is useless when the roots can be expressed without cube roots
☕ So, Let's solve this question by u r own by the formula
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☀️ Hence, solved !!
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