NCERT Solutions for Chapter: Polynomials, Exercise 1: Exercise

Shown below are the graphs of two cubic polynomials, $f\left(x\right)$ and $g\left(x\right)$. Both the polynomials have the zeroes $-1, 0$ and $1$

Anya said, " Both the graphs represent the same polynomial, $f\left(x\right)=g\left(x\right)=\left(x+1\right)\left(x-0\right)\left(x-1\right)$ as they have the same zeroes.

Pranit said," Both the graphs represent two different polynomial, $f\left(x\right)=\left(x+1\right)\left(x-0\right)\left(x-1\right)$ and $g\left(x\right)=-\left(x+1\right)\left(x-0\right)\left(x-1\right)$ and only two such polynomial exist that can have the zeroes $-1, 0$ and $1$

Aadar said, " Both the graphs represent two different polynomial and infintely many such polynomials exist that have zeroes $-1, 0$ and $1$.

Who is right and justify your answer?

Find the distance between the zeroes of the polynomial $f\left(x\right)=2x^2-x-6$. Show your steps.

$p\left(x\right)=2x^2-6x-3$. The two zeroes are of the form: $\frac{3\pm \sqrt{k}}{2}$; where $k$ is a real number.

Use the relationship between the zeroes and the coefficients of a polynomial to find the value of $k$. Show your steps.

Shown below are the parts of graphs of two polynomials, $g\left(x\right)$ and $h\left(x\right)$. When $h\left(x\right)$ is divided by $\left(x-3\right)$, the remainder is $k$

Which of these is true for the remainder when $g\left(x\right)$ is divided by $\left(x-3\right)$?

If $a(70a-12b+2c)>0$ and the quadratic equation $a{x}^2+bx+c=0$ real roots $\alpha$ and $\beta$, then which of the following statements is always false?

A graph of a cubic polynomial is plotted on a graph paper.

What is the sum of the roots of the polynomial?

The graph of polynomial $p\left(x\right)$ passes through the points $\left(-5, 0\right), \left(0, -40\right), \left(8, 0\right) \text{and} \left(5, -30\right)$. Which among the following is a factor of $p\left(x\right)$?

A polynomial is divided by $\left(x-1\right).$ The quotient obtained is $3x^3-x^2-x-4$ , and the remainder is $-5.$ Which polynomial meets these conditions?