Question

Polynomials are everywhere. It is found in the slope of a hill, the curve of a bridge or the continuity of a mountain range.

Based on the given information, answer the following questions.

(i) If the equation of the bridge is represented by the following graph $y=p\left(x\right)$, then name the type of the polynomial it traces.


(A) Linear
(B) Quadratic
(C) Cubic
(D) Bi-quadratic

(ii) If the path traced by the hills is represented by the graph $y=p\left(x\right)$ below, find the number of zeroes.


(A) 0
(B) 1
(C) 2
(D) 3​​​

(iii) Find a quadratic polynomial for the bridge if 6 is the sum and 8 is the product of its zeroes.

(A) $x^2+6x+8$
(B) $x^2-6x+8$
(C) $x^2+6x-8$
(D)​​​ $x^2-6x-8$

Answer 1

(i) Answer: (B) Quadratic

The graph here, cuts the x-axis at two different points.
Hence, the number of zeroes for the given graph is 2.

We know that, the degree of the polynomial indicates the maximum number of zeroes it can have.

Here, the maximum number of zeroes is 2.
Thus, the type of the polynomial it traces has to be a quadratic polynomial.
[1 mark]

(ii) Answer: (D) 3

In the given figure, the graph of a polynomial $p\left(x\right)$ cuts the x-axis at three distinct points. i.e the value of polynomial is equal to zero at these three points.

$\therefore$ The number of zeroes of $p\left(x\right)=3$.
[1 mark]

(iii) Answer: (B) $x^2-6x+8$

To find: A quadratic polynomial whose sum is 6 and the product is 8.

We know that, the general quadratic polynomial can be written as $x^2-\text{(Sum of zeroes)}x+\text{(Product of zeroes)}$

Given: Sum of zeroes = 6
Product of zeroes = 8

Hence, the required polynomial $=x^2-6x+8$
[1 mark]