10th Class Mathematics Polynomials

 

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

 

2. 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.

 

3. 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.

 

4. 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.

 

5. 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.

 

6. 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.

 

7. 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.

 

8. 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.

 

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

 

10. 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.

 

11. 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:

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

 

12. 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)

 

2. 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

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

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

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

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

 

3. 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\[~\alpha \beta \,\,=\text{ }35\].

 

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

 

4. 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:

            \[3{{k}^{2}}=\text{ }0\text{ }\Rightarrow \text{ }{{k}^{2}}=\text{ }0\text{ }\Rightarrow \text{ }k\text{ }=\text{ }0\]

 

5. 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.