Polynomials
Category : 10th Class
Polynomials
(a) A polynomial of degree 1 is called a linear polynomial.
(b) A polynomial of degree 2 is called a quadratic polynomial.
(c) A polynomial of degree 3 is called a cubic polynomial.
(d) A polynomial of degree 4 is called a biquadratic polynomial or a quadratic polynomial.
Polynomial 
General form 
Coefficients 
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 
\[a{{x}^{3}}+b{{x}^{2}}+cx+d\] 
\[a,b,c,d\in R,a\ne 0\] 
Quadratic polynomial 
\[a{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+e\] 
\[a,b,c,d,e\in R,a\ne 0\] 
In other words, a real number 'a' is called a zero of a polynomial p(x) if p(a) =0.
Note: The number zero is known as zero polynomial and its degree is not defined.
(a) The graph of a linear equation of the form \[y=ax+b,a\ne 0\]is a straight line which intersects the Xaxis at\[\left( \frac{b}{a},0 \right)\].
Zero of the polynomial ax + b is the xcoordinate of the point of intersection of the graph with Xaxis. Thus, the zero of\[y=ax+b\]is \[\left( \frac{b}{a} \right)\]
Note: A linear polynomial ax + b, a f 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 \[\cup \]when the coefficient of\[{{x}^{2}}\]is positive or opens downwards like\[\cap \]when the coefficient of\[{{x}^{2}}\]is negative.
The zeros of a quadratic polynomial\[a{{x}^{2}}+bx+c\]are the xcoordinates of the points where the parabola represented by\[y=a{{x}^{2}}+bx+c\]intersects the Xaxis.
(i) Case 1: if\[{{b}^{2}}4ac>0\]and\[a\ne 0\], then the graph of\[y=a{{x}^{2}}+bx+c\]intersects the Xaxis at two distinct points. The xcoordinates of the two points are the zeros of the quadratic polynomial\[a{{x}^{2}}+bx+c\].
(ii) Case2: lf \[{{\text{b}}^{\text{2}}}\text{4ac}=0\], and \[a\ne 0\],then the graph of \[y=a{{x}^{2}}+bx+c\]intersects the Xaxis at exactly one point (in fact at two coincident points).The xcoordinate of this point is the zero of the quadratic polynomial \[a{{x}^{2}}+bx+c\].
(iii) Case 3: If \[{{\text{b}}^{\text{2}}}\text{4ac}<0\], and \[a\ne 0\], then the graph of \[y=a{{x}^{2}}+bx+c\]does not intersect the Xaxis at any point. The graph is completely above or completely below the Xaxis. So, the quadratic polynomial \[a{{x}^{2}}+bx+c\]has no zero.
Thus, a quadratic polynomial can have either two distinct zeros, two equal zeros (i.e., one zero) or no zeros. Hence a polynomial of degree 2 has at most 2 zeros.
Note: For the parabola \[a{{x}^{2}}+bx+c\],
(i) Vertex \[=\left( \frac{b}{2a},\frac{D}{4a} \right)\]where \[D={{b}^{2}}4ac\].
(ii) Axis of symmetry, \[x=\frac{b}{2a}\]parallel to Yaxis.
(iii) Zeros are\[\frac{b+\sqrt{{{b}^{2}}4ac}}{2a}\]and\[\frac{b\sqrt{{{b}^{2}}4ac}}{2a}\].
(c) The graph of a cubic polynomial intersects the Xaxis at three points, whose x coordinates are the zeros of the cubic polynomial.
(d) In general, the graph of a polynomial y = p(x) passes through at most 'n' points on the Xaxis. Thus, a polynomial p(x) of degree 'n' has at most 'n' zeros.
Type of Polynomial 
General Form 
Number of Zeros 
Relationship between zeros and coefficients 

Sum of zeros 
Product of zeros 

Linear polynomial 
\[ax+b,\]\[a\ne 0\] 
1 
\[k=\frac{b}{a}\]i.e., \[k=\frac{(cons\tan t\,term)}{(coefficient\,of\,x)}\] 

Quadratic polynomial 
\[a{{x}^{2}}+bx+c\] \[a\ne 0\] 
2 
\[\frac{(coefficient\,of\,x)}{(cofficient\,of\,{{x}^{2}})}\]\[=\frac{b}{a}\] 
\[\frac{cons\tan t\,term}{coefficient\,of\,{{x}^{2}}}\] \[=\frac{c}{a}\] 
Cubic polynomial 
\[a{{x}^{3}}+b{{x}^{2}}+cx+d,\] \[a\ne 0\] 
3 
\[\frac{(coefficient\,of\,{{x}^{2}})}{(coefficient\,of\,{{x}^{2}})}\]\[=\frac{b}{a}\] 
\[\frac{cons\tan t\,term}{coefficient\,of\,{{x}^{2}}}\]\[=\frac{d}{a}\] 
Sum of the product of roots taken at a time \[\frac{coefficient\,of\,x}{coefficient\,of\,{{x}^{2}}}=\frac{c}{a}\] 
i.e., \[{{x}^{2}}\] (sum of the zeros) x + product of zeros.
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