Test: Geometric meaning of the Zeroes - Class 10 MCQ

Clarification: The graph of the polynomial 2x²-8x+5 cuts the y-axis.
Hence, the value of x will be 0.
y(0)=2(0)²-8(0)+5
y=5
The graph cuts the y-axis at (0,5)

We obtain zeros of the polynomial when y=0. Observing the points we see that we have point (5,0) which means when y=0 , x= 5 this means x=5 is a solution of the polynomial.

A cubic polynomial is a polynomial of the form . A cubic polynomial is a polynomial of degree 3. A univariate cubic polynomial has the form . An equation involving a cubic polynomial is called a cubic equation. A closed-form solution known as the cubic formula exists for the solutions of an arbitrary cubic equation.

A polynomial is a mathematical expression which is sum of several terms consisting the same variable in different powers. The degree of the polynomial is the highest power the variable is having in the expression. A quartic polynomial is a polynomial with degree 4. A polynomial with degree 4 can have 0, 1, 2, 3, or 4 roots. There are few methods developed to find the roots of a quartic equation, like Ferrari's solution, changing the equation to depressed form and then solve it or solving a quartic using radicals. These methods are little complex. A quartic function can have 0, 1 or 2 inflection points and have these inflection points implies the concept of golden ratio. 

The polynomial intersects the x-axis three times. A point on x-axis is of the form (a,0) where a is any number.So this means that any point on x axis has y coordinate equal to zero and we know that y=0 is when we have roots of the equation.So since there are three points at which y=0 so we have 3 roots of the equation. No. of roots is equal to the degree of the polynomial. So the degree of the polynomial is 3.

The graph of the polynomial
f (x) = 2x - 5

Method.

Let us consider, y = f (x) = 2x - 5

Now, when x = 0, y = - 5

and when y = 0, x = 5/2

So, we get two points (0, - 5) and (5/2, 0).

We plot the two points on a graph paper considering two rectangular axes XOX' and YOY'. Then, we find a straight line as shown in the graph.

We see that the line intersects x-axis at the point (5/2, 0).

The correct solution of this question is given below:

Here, P(a) = x²a² - 2xa + 3x - 2

1 is a zero of P(a), so P(1) = 0

Therefore, x²1² - 2x.1 + 3x - 2 = 0

x² + x - 2 = 0

(x + 2)(x - 1) = 0

x = -2, 1

Clarification: The graph of x²+2x+1 does not cut the x-axis, because it has imaginary roots.
x²+2x+2=0
D = b²-4ac = 2²-4(1)(2) = 4 - 8 = -4
as D<0 so it does not touch x-axis