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Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10 PDF Download

An algebraic identity is an equality that holds for any value of its variables. They are generally used in the factorization of polynomials or simplification of algebraic calculations. A polynomial is just a bunch of algebraic terms added together, for example, p(x) = 4x + 1 is a degree-1 polynomial. Similarly, polynomials can be of any degree 1, 2, 3… and so on. Polynomials are basically those Mathematical Expressions that make calculations easy in real life. There are different types of polynomials depending upon the number of terms present in them, for example, if there are 2 terms, it is known as a binomial.

Zeros of Polynomials

Let’s assume that P(x) is a polynomial. Let x = r, be the value of x where our polynomial P(x) becomes zero i.e,
P(x) = 0 at x = r or P(r) = 0

The process of finding the zeroes of P(x) is simply solving the equation P(x) = 0. We already know how to calculate zeros for first and second-degree polynomials. Let’s see some examples for this,

Question 1: P(x) = x2 + 2x – 15. Find the roots of this polynomial.
Solution: 
Let’s put P(x) = 0,
x2 + 2x -15 = 0
⇒ x2 + 5x – 3x – 15 = 0
⇒ x(x +5) -3 (x + 5) = 0
⇒ (x – 3) (x + 5) = 0
So, this expression will be zero for two values of x, i.e. x = 3 and -5. 

The last example used the factorization method for finding the roots of the polynomial. We can also use other methods such as the zero factor of the Shree Dharacharya Quadratic formula for finding out the roots or zeros of a polynomial.
If ax2 + bx + c = 0, where a≠0, then the formula for roots will be, 
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10
This formula for finding out roots of a polynomial is known as Shree Dharacharya Quadratic Formula. 

Question 2: Solve for the roots of the polynomial, x+ 6x – 14 = 0 
Solution: 
As it is clear that the above expression cannot be simplified just by intuition, therefore, we shall go for the Dharacharya Formula.
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10
Here, a=1, b=+6, c=-14 
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10
x=-3±√23 

Geometric Meaning of Zeros of a Polynomial

We know what zeros are, but why are they important? Let’s look at the graphical interpretation of the meaning behind zeros. Let’s look at the graphs of first and second-degree polynomials.  
Let’s take a polynomial y = 2x + 1, this is also an equation of a straight line. The figure below represents the graph of this straight line. We will look at the positions where y becomes zero.  
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10This graph intersects the x-axis at one position, there is only one root. So, we can conclude from the graph that roots are the position where the graph of polynomial intersects the x-axis.  
So, in general, for a polynomial y = ax + b, it represents a straight line with one root, which lies on the position where the graph cuts the x-axis i.e ( Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10
Now, let’s look at graphs of second-degree polynomials 
Let p(x) = x2 – 3x – 4, the figure below represents its graph. Let’s focus on the roots of this polynomial.  
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10In this graph, we can see that it cuts the x-axis at two points. So, these two points are the zeros of this polynomial. But will it be always this case? This graph can be upward facing or downward-facing depending upon the coefficient of x2 
In general, the zeroes of a quadratic polynomial ax2 + bx + c, a ≠ 0, are precisely the x-coordinates of the points where the parabola representing y = ax2 + bx + c intersects the x-axis. 
Three possible cases can happen to the shape of the graph.  
Case (i): The graph cuts the x-axis at two distinct points A and A′. This is the case that’s been shown above.  Here the polynomial y = ax2 + bx + c has two distinct roots.  
Case (ii): The graph cuts the x-axis at exactly one point, i.e., at two coincident points. Two same roots. Here the polynomial y = ax2 + bx + c has only one root.  
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10Case (iii): The graph is either completely above x-axis or completely below the x-axis. Here the polynomial y = ax2 + bx + c has no roots.  
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10Similarly, let’s study this for a cubic polynomial 

Cubic Polynomial

Let’s assume a polynomial, P(x) = x3 – 4x.  
P(x) = x(x2-4)   
= x(x-2)(x+2)
So the roots of this polynomial are x = 0, 2, -2.  
We can verify from the graph that these must be the places where the curve cuts the x-axis.  
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10But as in the case of second-degree polynomials, there may be more than one possibility.  
Let’s take an example of cubic polynomial P(x) = x3 
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10This graph cuts the x-axis only at one point. So, x = 0 is the only root of the polynomial p(x) = x3 
There can be another case, for example, take P(x) = x3 – x2
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10We can see that the graph of this function cuts the x-axis at two places only. From the above example, we can say that this polynomial can have at most 3 zeros.  

Note: In general a polynomial of degree “N” can have at most N zeros.  

Question 1: Which of the following graphs represents a cubic polynomial.  
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10
Answer:
(A) is the graph of a cubic polynomial because it cuts the x-axis only three times.
(B) does not cut the x-axis at all and is continuously increasing, So it cannot be.
(C) It cuts the x-axis at more than 5 points. So it cannot be three degree polynomial
(D) It is a graph of a parabola, we studied earlier. So it is not a cubic polynomial.

Question 2: Show the zeros of the Quadratic equation on the graph, x– 3x – 4 = 0 
Solution:
From the equation, we can tell that there are 2 values of x.
Factorizing the quadratic equation to find out the values of x,
x2-4x+x-4= 0
x(x-4) +1(x-4)= 0
x = (-1), x = 4
Hence, the graph will be an upward parabola intersecting at (-1,0) and (4,0)
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10


Question 3: Find the point where the graph intersects x-axis for the quadratic equation, x– 2x – 8 = 0 
Solution: 
Factorize the quadratic equation to find the points,
x2-4x+2x-8 = 0
x(x-4) +2(x-4)= 0
(x-4)(x+2)= 0
x = 4, x = (-2)
Therefore, the equation will cut the graph on x-axis at (4,0) and (-2,0)
Geometrical meaning of the Zeroes of a Polynomial | Mathematics (Maths) Class 10

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FAQs on Geometrical meaning of the Zeroes of a Polynomial - Mathematics (Maths) Class 10

1. What is the geometrical meaning of the zeroes of a polynomial?
Ans. The geometric meaning of the zeroes of a polynomial refers to the points on the graph of the polynomial where the function intersects or crosses the x-axis. These points represent the values of x for which the polynomial function evaluates to zero.
2. How can the zeroes of a polynomial be determined geometrically?
Ans. To determine the zeroes of a polynomial geometrically, we plot the graph of the polynomial function on a coordinate plane. The points where the graph intersects the x-axis are the x-values for which the polynomial evaluates to zero, indicating the zeroes of the polynomial.
3. What does it mean when a polynomial has no zeroes?
Ans. When a polynomial has no zeroes, it means that the graph of the polynomial function does not intersect or cross the x-axis. In other words, there are no values of x for which the polynomial evaluates to zero. This can occur when the graph is entirely above or below the x-axis.
4. Can a polynomial have more than one zero?
Ans. Yes, a polynomial can have more than one zero. In fact, the fundamental theorem of algebra states that a polynomial of degree n will have exactly n complex zeroes, counting multiplicity. This means that a polynomial can have multiple zeroes, including repeated ones.
5. How does the multiplicity of a zero affect the graph of a polynomial?
Ans. The multiplicity of a zero affects the graph of a polynomial by determining how the graph interacts with the x-axis at that point. If a zero has an odd multiplicity, the graph will cross the x-axis at that point. If the multiplicity is even, the graph will touch the x-axis but not cross it. Additionally, the steepness or flatness of the graph near the zero is also influenced by the multiplicity.
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