# 10th Class Mathematics Polynomials

Polynomials

Category : 10th Class

Polynomials

Polynomials

1. Polynomials: If x is a variable, n be a positive integer and ${{a}_{0}},\text{ }{{a}_{1}},\text{ }{{a}_{2}}\ldots .,\text{ }{{a}_{n}}$ are real number, then an expression of the form $p\left( x \right)\text{ }=\text{ }{{a}_{0}}+\text{ }{{a}_{1}}x\text{ }+\text{ }{{a}_{2}}{{x}^{2}}+\text{ }{{a}_{n}}{{x}^{n}}$ is called polynomial, in the variable x. In a polynomial, $p\left( x \right)\text{ }=\text{ }{{a}_{0}}+\text{ }{{a}_{1}}x+\text{ }{{a}_{2}}{{x}^{2}}+\text{ }\ldots \ldots .\text{ }+\text{ }{{a}_{n}}{{x}^{n}},\text{ }{{a}_{0}},\text{ }{{a}_{1}}x,\text{ }{{a}_{2}}{{x}^{2}},\ldots .,\text{ }{{a}_{n}}{{x}^{2}}$ are known as the terms of the polynomial and       ${{a}_{0}},\,\,{{a}_{1}},\text{ }{{a}_{2}}\ldots \ldots ..$, an are known as their coefficients

1. Degree of a polynomial: Let p(x) be a polynomial in x. Then, the highest power of x in p(x) is called the degree of the polynomial p(x). Thus, the degree of the polynomial, $p\left( x \right)\text{ }=\text{ }{{a}_{0}}+\text{ }{{a}_{1}}x\text{ }+\text{ }{{a}_{2}}{{x}^{2}}+\text{ }\ldots \ldots .\text{ }+\text{ }{{a}_{n}}{{x}^{n}}$, where an $\ne$ 0 is n.

1. Constant polynomial: A polynomial of degree zero is called a constant polynomial e.g. $p\left( x \right)\text{ }=\text{ }-\text{ }5$ is a constant polynomial.

1. Zero polynomial: The constant polynomial $p\left( x \right)\text{ }=\text{ }0$ is called the zero polynomial. The degree of the zero polynomial is not defined since $p\left( x \right)\,\,=\,\,0\,\,=\,\,0.x\,\,=\,\,0.{{x}^{2}}\,\,=\,\,0.{{x}^{3}}\,\,=$… etc.

1. Linear polynomial: A polynomial of degree 1 is called a linear polynomial A linear polynomial is of the form             $p\left( x \right)~~=\text{ }ax\text{ }+\text{ }b$, where $a~\,\,\ne \,\,0$ e.g. $5x\text{ }+\text{ }1$, $-\frac{5}{2}x,\,\,2\sqrt{3}x-\sqrt{2}\,\,etc.$ are linear polynomials.

1. Quadratic polynomial: A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial is of the form $p\left( x \right)\text{ }=\text{ }a{{x}^{2}}+\text{ }bx\text{ }+\text{ }c$, where$a~\,\,\ne \,\,0$.

e.g.${{x}^{2}}-5,\,\,\,5\sqrt{2}{{x}^{2}}-\frac{1}{\sqrt{3}}x,\,\,7{{x}^{2}}\,\,+\,\,\sqrt{5}$ etc. are quadratic polynomials.

1. Cubic polynomial: A polynomial of degree 3 is called a cubic polynomial A cubic polynomial is of the form $p\left( x \right)~~=\text{ }a{{x}^{2}}+\text{ }b{{x}^{2}}+\text{ }cx\text{ }+\text{ }d$, where$a~\,\,\ne \,\,0$.

e.g. ${{x}^{3}}-20,\,\,\sqrt{5}{{x}^{3}}\,\,-\,\,\frac{1}{9}x,\,\,\frac{7}{2}{{x}^{3}}-\frac{1}{2}{{x}^{2}}-4$etc. are cubic polynomials.

1. Biquadratic polynomial: A polynomial of degree 4 is called a biquadratic polynomial. A biquadratic polynomial is of the form$p\left( x \right)\text{ }=\text{ }a{{x}^{4}}+\text{ }b{{x}^{3}}+\text{ }c{{x}^{2}}dx\text{ }+\text{ }e$, where $a~\,\,\ne \,\,0$.

e.g. ${{x}^{4}}\text{ }23,\,\sqrt{3}{{x}^{4}}-\frac{1}{9}x,\,\,\frac{1}{2}{{x}^{4}}+\frac{3}{4}x-\frac{1}{8}$ etc. are biquadratic polynomials.

1. Zeros of a polynomial: A real number k is said to be a zero of the polynomial p(x), if$p\left( k \right)\text{ }=\text{ }0$.

1. Relationship between the Zeros and Coefficients of a Linear Polynomial: The zero of a linear polynomial

$p\left( x \right)\text{ }=\text{ }ax\text{ }+\text{ }b$ is given by$\alpha =\frac{-b}{a}=\frac{-(constant\,\,term)}{(coefficient\,\,of\,\,x)}$.

A linear polynomial can have at the most one zero.

1. Relationship between the Zeros and Coefficients of a Quadratic Polynomial:

(i) If $\alpha$ and $\beta$ are the zeros of a quadratic polynomial $p\left( x \right)\text{ }=\text{ }a{{x}^{2}}+\text{ }bx\text{ }+\text{ }c,\text{ }a\,\,\ne \,~0$ then

$\alpha +\beta \,\,=\,\,\frac{-b}{a}\,\,=\,\,\frac{-(coefficient\,\,of\,\,x)}{(coefficien\,\,of\,\,{{x}^{2}})};$

(ii)  A quadratic polynomial whose zeroes are $\alpha$ and $\beta$ is given by:

$p\left( x \right)\text{ }=\text{ }{{x}^{2}}\,-\,\,(\alpha +\beta )\,\,+\text{ }(\alpha \beta )$

1. Relationship between the Zeros and Coefficients of a Cubic Polynomial:

(i) If $\alpha ,\,\,\beta \,\,and\,\,\gamma$ are the zeros of$p\left( x \right)\text{ }=\text{ }a{{x}^{3}}+\text{ }b{{x}^{2}}+\text{ }cx\text{ }+\text{ }d$, then

$(\alpha \,\,+\,\,\beta \,\,+\,\,\gamma )\,\,=\,\,\frac{-b}{a}\,\,=\,\,\frac{-(coefficient\,\,of\,\,{{x}^{2}})}{(coefficient\,\,of\,\,{{x}^{3}})};$

$(\alpha \beta \,\,+\,\,\beta \gamma \,\,+\,\,\gamma \alpha )\,\,=\,\,\frac{c}{a}\,\,=\,\,\frac{(coefficient\,\,of\,\,x)}{(coefficient\,\,of\,\,{{x}^{3}})};$

$\alpha \beta \gamma \,\,=\,\,\frac{-d}{a}\,\,=\,\,\frac{-(cons\tan t\,\,term)}{(coefficient\,\,of\,\,{{x}^{3}})}$

(ii)    A cubic polynomial whose zeros are $\alpha ,\,\,\beta \,\,and\,\,\gamma$ is given by

$p(x)=\{{{x}^{3}}-(\alpha +\beta +\gamma ){{x}^{2}}+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma \}$

Snap Test

1. A real number a is called a zero of the polynomial f(x) if

(a) $f\left( 0 \right)\text{ }=\text{ }a$

(b) $f\left( a \right)\text{ }-\text{ }a$

(c)$f\left( a \right)=0$

(d) $f\left( a \right)=f\left( 0 \right)$

(e) None of these

Ans.     (c)

Explanation: We know that for the polynomial f(x), if f$\left( a \right)\text{ }=\text{ }0$, then a is a zero of the polynomial f(x)

1. If zeros of a quadratic polynomial are $\mathbf{(-3+}\sqrt{\mathbf{3}}\mathbf{)}\,\,\mathbf{and}\,\,\mathbf{(-3-}\sqrt{\mathbf{3}}\mathbf{)}$, find the polynomial.

(a) ${{x}^{2}}+\text{ }6x\text{ }+\text{ }6$

(b) ${{x}^{2}}-\text{ }6x\text{ }+\text{ }6$

(c) ${{x}^{2}}+\text{ }2x\text{ }+\text{ }4$

(d) $x\text{ }+\text{ }6x\text{ }-\text{ }6$

(e) None of these

Ans.     (a)

Explanation: Required polynomial

$f(x)\,\,\,\,=\,\,\,\,[x-(-\,3\,\,+\,\,\sqrt{3})]\,\,[x-(-\,\,3\,\,-\sqrt{3})]$

=  $[(x+3)-\sqrt{3}]\,\,\,[(x+3)+\sqrt{3}]$

=  ${{(x+3)}^{2}}-{{(\sqrt{3})}^{2}}$         $[\,\,\because \,\,\,~\left( a\text{ }-\text{ }b \right)\left( a\text{ }+\text{ }b \right)\text{ }=\text{ }{{a}^{2}}-\text{ }{{b}^{2}}]$

$\,=\,\,\,\,\,\left( {{x}^{2}}+\text{ }6x\text{ }+\text{ }9 \right)\text{ }-\text{ }3\text{ }=\text{ }{{x}^{2}}+\text{ }6x\text{ }+\text{ }6$.

1. If $\alpha \,\mathbf{and}\,\beta$ are the zeros of the polynomial $\mathbf{p}\left( \mathbf{x} \right)=\text{ }{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{12x}\text{ }+\text{ }\mathbf{35}$, evaluate $\frac{\mathbf{1}}{\alpha }\mathbf{+}\frac{\mathbf{1}}{\beta }$.

(a) $\frac{-10}{35}$     (b) $\frac{-12}{35}$

(c) $\frac{-14}{35}$      (d) $\frac{-11}{35}$

(e) None of these

Ans.     (b)

Explanation: Given that $~\alpha \,and\,\beta$ are the zeros of the polynomial $p\left( x \right)=\text{ }{{x}^{2}}+\text{ }12x\text{ }+\text{ }35$, Therefore, $~\alpha +\beta \,\,~=-\text{ }12$ and$~\alpha \beta \,\,=\text{ }35$.

$\therefore \,\,\,\frac{1}{\alpha }\,\,+\,\,\frac{1}{\beta }\,\,=\,\,\frac{\alpha +\beta }{\alpha \beta }\,\,=\,\,\frac{-12}{35}$

1. One of the zeros of the quadratic polynomial $\mathbf{f}\left( \mathbf{x} \right)=\text{ }\mathbf{14}{{\mathbf{x}}^{\mathbf{2}}}-\text{ }\mathbf{42}{{\mathbf{k}}^{\mathbf{2}}}\mathbf{x}-\text{ }\mathbf{9}$ is negative of the other. Find the value of k.

(a) $k\text{ }=\text{ }0$

(b) $k\text{ }=\text{ }1$

(c) $k\text{ }=\text{ }3$

(d) $k\text{ }=\text{ }2$

(e) None of these

Ans.     (a)

Explanation: Let $~\alpha \,\,and\,\,\beta$ be the zeros of the polynomial, $f\left( x \right)=\text{ }14{{x}^{2}}-\text{ }42\text{ }{{k}^{2}}x\text{ }-\text{ }9$.

Then     $\alpha +\beta \,\,\,\,\,\,\,=\,\,\,\,\,\frac{42{{k}^{2}}}{14}\,\,\,\,=\,\,\,\,\,\,3{{k}^{2}}$      [Sum of the zeros of f(x)]

Now,  let    $\beta \,\,\,\,\,\,\,\,=\,\,\,\,\,\,\,(-\alpha )$   [Since one of the zeros of f(x) is negative of the other]

Then, $~~\alpha \,\,+\,\,\beta ~\,\,=\text{ }0$.

Equating the two values of $(\alpha +\beta )$ we get:

$3{{k}^{2}}=\text{ }0\text{ }\Rightarrow \text{ }{{k}^{2}}=\text{ }0\text{ }\Rightarrow \text{ }k\text{ }=\text{ }0$

1. The zeros of the cubic polynomial $\mathbf{f}\left( \mathbf{x} \right)\,\,=\text{ }{{\mathbf{x}}^{\mathbf{3}}}-\text{ }\mathbf{6}{{\mathbf{x}}^{\mathbf{2}}}-\text{ }\mathbf{13x}\text{ }+\text{ }\mathbf{42}$ are in A.P. Find the its zeros.

(a) -2, 3 and 7                (b) -1, 2 and 5

(c) -3, 2 and 7               (d) -4, 2 and 5

(e) None of these

Ans.     (c)

Explanation: Let $(\alpha -d),\alpha \,\,and\,(\alpha +d)$ be the zeros of the polynomial

$f\left( x \right)\text{ }=\text{ }{{x}^{3}}-\text{ }6{{x}^{2}}-\text{ }13x\text{ }+\text{ }42$                  (Since the zeros are in A.P.)

Then,    Sum of the zeros of $f\left( x \right)\text{ }=\text{ }6$

i.e. $\left( \alpha \text{ }-\text{ }d \right)\text{ }+\text{ }\alpha \text{ }+\text{ }\left( \alpha \text{ }+\text{ }d \right)\text{ }=\text{ }6\text{ }\Rightarrow \text{ }3\alpha \text{ }=\,\,\,6\,\,\,\Rightarrow \text{ }\alpha \,\,=\text{ }2$

Also,       Product of the zeros of $f\left( x \right)\text{ }=\text{ }-\text{ }42$

i.e. $\left( \alpha \text{ }-\text{ }d \right)\text{ }\alpha \left( \alpha \text{ }+\text{ }d \right)\text{ }=\text{ }-\text{ }42$

$\Rightarrow \,\,\,\,\alpha \left( {{\alpha }^{2}}-{{d}^{2}} \right)=-\,42\,\,\Rightarrow \,\,2\left( {{2}^{2}}-{{d}^{2}} \right)\,\,=-\,42$

$\Rightarrow \,\,{{d}^{2}}=\text{ }25\text{ }\Rightarrow \text{ }d\,\,=\pm \text{ }5$

Taking any of the values of d i.e. taking either $d\,\,=\,\,5$ or $d\text{ }=\text{ }-\text{ }5$, we get the zeros of f (x) as - 3, 2 and 7.

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