# 10th Class Mathematics Polynomials Polynomial

Polynomial

Category : 10th Class

POLYNOMIAL

FUNDAMENTALS

• Polynomial: A function p(x) of the form $p(x)={{a}_{0}}+{{a}_{1}}{{x}^{n}}+......+{{a}_{n}}{{x}^{n}},$where ${{a}_{0}},{{a}_{1}},.......{{a}_{n}}$an are real numbers and ‘n’ is a non-negative (positive) integer is called a polynomial.

Note: ${{\mathbf{a}}_{\mathbf{0}}}\mathbf{,}{{\mathbf{a}}_{\mathbf{1}}}\mathbf{,}.....{{\mathbf{a}}_{\mathbf{n}}}$ are called the coefficients of the polynomial.

• If the coefficients of a polynomial are integers, then it is called a polynomial over integers.
• If the coefficients of a polynomial are rational numbers, then it is called a polynomial over rational numbers.
• If the coefficients of a polynomial are real numbers, then it is called a polynomial over real numbers.
• A function $p(x)={{a}_{0}}+{{a}_{1}}x+........+{{a}_{n}}{{x}^{n}}$ is not a polynomial if the power of the variable is either negative or a fractional number.
• Standard form: A polynomial is said to be in a standard form if it is written either in the ascending or descending powers of the variable, as $1+x+2{{x}^{2}}+3{{x}^{3}}-6{{x}^{5}}\times 6{{x}^{6}}$
• Degree of a polynomial: The highest power of x in p(x) is the degree 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 $a{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+c$ $a,b,c,d,e\in R,a\ne 0$

• Value of a polynomial: If p(x) is a polynomial in x, and if ‘a’ is any real number, then the value obtained upon replacing ‘x’ by ‘a’ in p(x) is denoted as p(a).
• Zero of a polynomial: A real number ‘a’ for which the value of the polynomial p(x) is zero, is called the zero of the polynomial.
• In other words, a real number ‘a’ is called a zero of a polynomial p(x) if p(a) = 0.

• Geometric meaning of the zero of a polynomial:
• The graph of a linear equation of the form $y=ax+b,a\ne 0$ is a straight line which intersects the X-axis at$\left( \frac{-b}{a},0 \right)$

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.

• Relationship between zeros and coefficients of a polynomial.
 Types of Polynomial General Form Number of Zeroes Relationship between zeroes and coefficients Sum of zeroes Product of zeroes 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 $\frac{coefficient\,of\,x}{coefficient\,of\,{{x}^{2}}}=\frac{c}{a}$.
• To form a quadratic polynomial with the given zeros: If $\alpha$ and $\beta$ are the zeros of a quadratic polynomial, then the quadratic polynomial is obtained by expanding ($x-\alpha$) ($x-\beta$). i.e., $(x-\alpha )(x-\beta )={{x}^{2}}-$(Sum of the zeros) x + product of zeros.
• To form a cubic polynomial with the given zeros: If $\alpha ,\beta$ and $\gamma$ are the zeros of a polynomial, then the cubic polynomial is obtained by expanding $(x-\alpha )(x-\beta )(x-\gamma )$.
• Division algorithm of polynomials: If $p\left( x \right)$and $g\left( x \right)$are any two polynomials with$g(x)\ne 0$, then we can find polynomials $q\left( x \right)$and $r\left( x \right)$such that$p\left( x \right)=\text{ }g\left( x \right)\times q\left( x \right)+r\left( x \right)$, where either $r(x)=0$ or degree $r\left( x \right)<$degree of$g\left( x \right)$.
• If $\left( x-a \right)$ is a factor of polynomial $p\left( x \right)$of degree$n>0$, then ‘a’ is the zero of the polynomial.

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 When 'a' is negative i.e. a < 0

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