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Q4(v):
Find the zero of the polynomial in each of the following cases: (v) $p(x) = 3x$

Solution :

Given Polynomial & Theoretical Foundation

We are given the linear polynomial:

$p(x) = 3x$

Theoretical Definition: The "zero" or "root" of a polynomial $p(x)$ is defined as any real or complex number $c$ such that when $x$ is replaced by $c$, the value of the polynomial evaluates to zero. Mathematically, $c$ is a zero if and only if $p(c) = 0$ [Per the Fundamental Theorem of Algebra and the Factor Theorem].

Step 1: Setting up the Equation

To find the zero of the polynomial, we must equate the polynomial expression to zero. This transforms our polynomial expression into an algebraic equation.

$p(x) = 0$

Substituting the given expression for $p(x)$:

$3x = 0$

Step 2: Algebraic Manipulation

We now solve for the variable $x$. The term $3x$ represents the product of the constant $3$ and the variable $x$.

To isolate $x$, we apply the Multiplicative Property of Equality, dividing both sides of the equation by the coefficient $3$:

$\frac{3x}{3} = \frac{0}{3}$

Since any non-zero number dividing zero results in zero [Per the Zero Product Property and properties of real numbers]:

$x = 0$

Step 3: Verification of the Root

To ensure absolute mathematical rigor, we verify the solution by substituting $x = 0$ back into the original polynomial $p(x)$:

$p(0) = 3(0)$

$p(0) = 0$

Because the polynomial evaluates to $0$ when $x = 0$, the solution is verified as correct.

Graphical Representation of the Polynomial

Geometrically, the zero of a polynomial with real coefficients corresponds to the $x$-coordinate of the point where the graph of the function $y = p(x)$ intersects the $x$-axis. For $y = 3x$, this is a straight line passing through the origin.

x y 3 -3 p(x) = 3x Zero: (0, 0)

As demonstrated in the Cartesian plane above, the line $y = 3x$ intersects the $x$-axis exactly at the origin $(0,0)$, visually confirming that the zero of the polynomial is $0$.

Final Conclusion

Final Solution: The zero of the polynomial $p(x) = 3x$ is $x = 0$.


More Questions from Class 9 Mathematics Polynomials EXERCISE 2.2


CBSE Solutions for Class 9 Mathematics Polynomials


Chapters in CBSE - Class 9 Mathematics


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