Find the number of polynomials having zeroes as $-2$ and $5$.
Given: Zeroes of a polynomial are $-2$ and $5$.
To do: To find the number of polynomials having zeroes as $-2$ and $5$.
Solution:
Let $\alpha=-2$ and $\beta=5$
$\therefore \alpha+\beta=-2+5=3\ ......( i)$
$\alpha\beta=-2\times5=-10\ .......( ii)$
$\therefore$ The polynomial $f( x)=x^2-( \alpha+\beta)x+\alpha\beta$
$\Rightarrow f( x)=x^2-3x-10$
If we multiply $f( x)$ by any constant $k$, roots will be $\alpha$ and $\beta$.
$\Rightarrow P( x)=k( x^2-3x-10)$
Thus, there are infinite number of polynomials possible.
Related Articles
- Choose the correct answer from the given four options in the following questions:The number of polynomials having zeroes as \( -2 \) and 5 is(A) 1(B) 2(C) 3(D) more than 3
- Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:\( 4 x^{2}-3 x-1 \)
- Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients:$8x^2-22x-21$.
- Find the quadratic polynomial whose zeroes are $-2$ and $-5$.
- Given that is $x-\sqrt{5}$ a factor of the polynomial $x^{3} -3\sqrt{5} x^{2} -5x+15\sqrt{5}$ , find all the zeroes of the polynomials.
- Find \( k \) so that \( x^{2}+2 x+k \) is a factor of \( 2 x^{4}+x^{3}-14 x^{2}+5 x+6 \). Also find all the zeroes of the two polynomials.
- Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:$3x - 2$
- Find \( \mathrm{k} \) so that \( x^{2}+2 x+k \) is a factor of \( 2 x^{4}+x^{3}-14 x^{2}+5 x+6 \). Also, find all the zeroes of the two polynomials.
- Classify the following polynomials as polynomials in one-variable, two-variables etc.$t^3 -3t^2 + 4t-5$
- Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:$2x + x^2$
- Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:$t^2 + 1$
- For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.\( \frac{-8}{3}, \frac{4}{3} \)
- If $\alpha$ and $\beta$ are the zeroes of the polynomial $f(x)=x^2−px+q$, then write the polynomial having $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ as its zeroes.
- Find the number of zeroes using C++
- Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:$x + x^2 + 4$
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies