Question

what do you mean by cubic polynomial​

Answer 1

Answer:

A cubic polynomial is a polynomial of degree 3. A univariate cubic polynomial has the form. . An equation involving a cubic polynomial is called a cubic equation. A closed-form solution known as the cubic formula exists for the solutions of an arbitrary cubic equation.

Step-by-step explanation:

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Answer 2

cubic function is a cubic curve, though many cubic curves are not graphs of functions.

Although cubic functions depend on four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always similar to the graph of a function of the form

{\displaystyle y=x^{3}+px.}{\displaystyle y=x^{3}+px.}

This similarity can be built as the composition of translations parallel to the coordinates axes, a homothecy (uniform scaling), and, possibly, a reflection (mirror image) with respect to the y-axis. A further non-uniform scaling can transform the graph into the graph of one among the three cubic functions

{\displaystyle {\begin{aligned}y&=x^{3}+x\\y&=x^{3}\\y&=x^{3}-x.\end{aligned}}}{\displaystyle {\begin{aligned}y&=x^{3}+x\\y&=x^{3}\\y&=x^{3}-x.\end{aligned}}}

This means that there are only three graphs of cubic functions up to an affine transformation.

The above geometric transformations can be built in the following way, when starting from a general cubic function {\displaystyle y=ax^{3}+bx^{2}+cx+d.}{\displaystyle y=ax^{3}+bx^{2}+cx+d.}

Firstly, if a < 0, the change of variable x → –x allows supposing a > 0. After this change of variable, the new graph is the mirror image of the previous one, with respect of the y-axis.

Then, the change of variable x = x1 –

b

/

3a

provides a function of the form

{\displaystyle y=ax_{1}^{3}+px_{1}+q.}{\displaystyle y=ax_{1}^{3}+px_{1}+q.}

This corresponds to a translation parallel to the x-axis.

The change of variable y = y1 + q corresponds to a translation with respect to the y-axis, and gives a function of the form

{\displaystyle y_{1}=ax_{1}^{3}+px_{1}.}{\displaystyle y_{1}=ax_{1}^{3}+px_{1}.}

The change of variable {\displaystyle \textstyle x_{1}={\frac {x_{2}}{\sqrt {a}}},y_{1}={\frac {y_{2}}{\sqrt {a}}}}{\displaystyle \textstyle x_{1}={\frac {x_{2}}{\sqrt {a}}},y_{1}={\frac {y_{2}}{\sqrt {a}}}} corresponds to a uniform scaling, and give, after multiplication by {\displaystyle {\sqrt {a}},}{\displaystyle {\sqrt {a}},} a function of the form

{\displaystyle y_{2}=x_{2}^{3}+px_{2},}{\displaystyle y_{2}=x_{2}^{3}+px_{2},}

which is the simplest form that can be obtained by a similarity.

Then, if p ≠ 0, the non-uniform scaling {\displaystyle \textstyle x_{2}=x_{3}{\sqrt {|p|}},\quad y_{2}=y_{3}{\sqrt {|p|^{3}}}}{\displaystyle \textstyle x_{2}=x_{3}{\sqrt {|p|}},\quad y_{2}=y_{3}{\sqrt {|p|^{3}}}} gives, after division by {\displaystyle \textstyle {\sqrt {|p|^{3}}},}{\displaystyle \textstyle {\sqrt {|p|^{3}}},}

{\displaystyle y_{3}=x_{3}^{3}+x_{3}\operatorname {sign} (p),}{\displaystyle y_{3}=x_{3}^{3}+x_{3}\operatorname {sign} (p),}

where {\displaystyle \operatorname {sign} (p)}{\displaystyle \operatorname {sign} (p)} has the value 1 or –1, depending on the sign of p. If one defines {\displaystyle \operatorname {sign} (0)=0,}{\displaystyle \operatorname {sign} (0)=0,} the latter form of the function applies to all cases (with {\displaystyle x_{2}=x_{3}}{\displaystyle x_{2}=x_{3}} and {\displaystyle y_{2}=y_{3}}{\displaystyle y_{2}=y_{3}}).