Polynomial
Category : 10th Class
POLYNOMIAL
FUNDAMENTALS
Note: \[{{\mathbf{a}}_{\mathbf{0}}}\mathbf{,}{{\mathbf{a}}_{\mathbf{1}}}\mathbf{,}.....{{\mathbf{a}}_{\mathbf{n}}}\] are called the coefficients of the polynomial.
Example: \[2-3{{x}^{5}}+6{{x}^{4}}+92{{x}^{3}}\]: Here, highest term being \[-3{{x}^{5}}\]: degree of polynomial = 5.
Polynomial |
General Form |
Coefficients |
Zero polynomial |
\[0\] |
\[-\] |
Linear polynomial |
\[ax+b\] |
\[a,b\in R,a\ne 0\] |
Quadratic polynomial |
\[a{{x}^{2}}+bx+c\] |
\[a,b,c\in R,a\ne 0\] |
Cubic polynomial |
\[a{{x}^{3}}+b{{x}^{2}}+cx+d\] |
\[a,b,c,d\in R,a\ne 0\] |
Bi-Quadratic polynomial |
\[a{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+c\] |
\[a,b,c,d,e\in R,a\ne 0\] |
Zero of the polynomial \[ax+b\]is the x-coordinate of the point of intersection of the graph with X-axis.
Note: A linear polynomial \[ax+b,a\ne 0\]has exactly one zero, i.e., \[\left( \frac{-b}{a} \right)\]
(b) The graph of a quadratic equation \[y=a{{x}^{2}}+bx+c,a\ne 0\]is a curve called parabola that either opens upwards like when the coefficient of\[{{x}^{2}}\] is positive or opens downwards like when the coefficient of\[{{x}^{2}}\] is negative.
The zeros of a quadratic polynomial \[a{{x}^{2}}+bx+c\]are the x-coordinates of the points where the parabola intersects the X-axis.
Example: \[p(x)={{x}^{2}}+2x+4=0;\] \[{{b}^{2}}-4ac\text{ }={{2}^{2}}-4.1.4=-12<0\]
It has no zeros as the parabola will never intersect X-axis.
Note: For the parabola \[a{{x}^{2}}+bx+c.\]
(i) Vertex \[\left( \frac{\mathbf{-b}}{\mathbf{2a,}}\mathbf{-}\frac{\mathbf{D}}{\mathbf{4a}} \right)\]where \[\mathbf{D=}{{\mathbf{b}}^{\mathbf{2}}}\mathbf{-4ac}\]
(ii) Axis of symmetry, \[\mathbf{x=}\frac{\mathbf{-b}}{\mathbf{2a}}\] parallel to Y-axis.
(ii) Zeros are\[\frac{\mathbf{-b+}\sqrt{{{\mathbf{b}}^{\mathbf{2}}}\mathbf{-4ac}}}{\mathbf{2a}}\]and \[\frac{\mathbf{-b-}\sqrt{{{\mathbf{b}}^{\mathbf{2}}}\mathbf{-4ac}}}{\mathbf{2a}}\].
(c) The graph of a cubic polynomial intersects the X-axis at three points, whose x-coordinates are the zeros of the cubic polynomial.
In general, the graph of a polynomial of degree ‘n’ y = p(x) passes through at most ‘n’ points on the X-axis. Thus, a polynomial p(x) of degree ‘n’ has at most ‘n’ zeros.
Types of Polynomial |
General Form |
Number of Zeroes |
Relationship between zeroes and coefficients |
|
Sum of zeroes |
Product of zeroes |
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Linear Polynomial |
\[ax+b,\] \[a\ne 0\] |
1 |
\[only\,\,one\,\,zero=\frac{-(constant\,\,term)}{(coefficient\,\,of\,\,x)}=\frac{-b}{a}\] |
|
Quadratic Polynomial |
\[a{{x}^{2}}+bx+c,a\ne 0\] |
2 |
\[\frac{-(coefficient\,\,of\,\,x)}{(coefficient\,\,of\,\,{{x}^{2}})}=\frac{-b}{a}\] |
\[\frac{constant\,\,term}{coefficient\,\,of\,\,{{x}^{2}}}=\frac{c}{a}\] |
Cubic Polynomial |
\[a{{x}^{3}}+b{{x}^{2}}+cx,+d,a\ne 0\] |
3 |
\[\frac{-(coefficient\,\,of\,\,x)}{(coefficient\,\,of\,\,{{x}^{2}})}=\frac{-b}{a}\] |
\[\frac{cons\tan t\,\,term}{coefficient\,\,of\,\,{{x}^{2}}}=\frac{-d}{a}\] |
Sum of the product of roots taken two at a time \[\frac{coefficient\,of\,x}{coefficient\,of\,{{x}^{2}}}=\frac{c}{a}\]. |
Graphical Representation of Different forms of Quadratic Equation
Characteristics of the function |
\[{{\mathbf{b}}^{\mathbf{2}}}\mathbf{-4ac<0}\] |
\[{{\mathbf{b}}^{\mathbf{2}}}\mathbf{-4ac=0}\] |
\[{{\mathbf{b}}^{\mathbf{2}}}\mathbf{-4ac>0}\] |
When 'a' is positive i.e. a > 0 |
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When 'a' is negative i.e. a < 0 |
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