Zero Polynomial

A polynomial is an algebraic phrase with one or more terms, as we are previously aware. The real values of the variable for which the value of the polynomial becomes 0 are known as polynomial zeroes. Therefore, if $p(m)=0$ and $p(n)=0$, then the real numbers 'm' and 'n' are zeroes of the polynomial p(x). A polynomial with the value zero (0) is referred to as a zero polynomial. The greatest power of the variable x in the polynomial ${ax}^2+bx+c=0$ is a polynomial's degree. A degree 1 polynomial is referred to as a linear polynomial.

What are Zero Polynomials?

Any real value of x for which the polynomial's value becomes 0 is defined as the polynomial's zero. If p(k) = 0, then a real integer k is the zero of the polynomial p(x).

Geometrical Meaning of the Zero Polynomial

The x-coordinate of the place where the graph intersects the x-axis serves as the polynomial zero. When a polynomial $p(x$) meets the x-axis at the coordinates $(k,0),k$ is the polynomial's zero.

• At most one point, the graph of a linear polynomial crosses the x-axis.

• A quadratic polynomial graph can intersect the x-axis up to two times. The graph in this instance has a parabola-like form.

• A quadratic polynomial might contain two separate zeros, two equal zeroes, or no zero geometrically.

• A cubic polynomial graph can cross the x-axis a maximum of three times. There can be a maximum of three zeros in a cubic polynomial.

• An nth-degree polynomial typically crosses the x-axis a maximum of $n$ times. A polynomial of the nth degree can only have n zeroes at most.

Degree of a Polynomial

The degree of a polynomial is determined by the variable term's highest exponential power. Let’s discuss some types of polynomials based on degree:

• A linear polynomial is a polynomial with a degree of $1$. $ax+b$, where $a$ and $b$ are real numbers and are not equal $0$. A linear polynomial is $2x+3$.

• A degree two polynomial is referred to as a quadratic polynomial. A quadratic polynomial has the standard form of $a{x}^2+bx+c$, where $a,b$, and $c$ are All real numbers, and a not equal zero, $x^2+3x+4$ is an example.

• A cubic polynomial is a three-degree polynomial. The formula for standard form is $a{x}^3+b{x}^2+cx+d$, where $a,b,c$, and $d$ are all real integers and not equal to zero. An illustration $x^3+x^2$.

Representing Zero Polynomial on Graph

A graph spanning the coordinate axis can show a polynomial expression of the form $y=f(x)$. On the x-axis is displayed the value of $x$, and on the y-axis is displayed the value of $f(x)$ or $y$. Depending on the degree of the polynomials, the polynomial expression may take the form of a linear expression, quadratic expression, or cubic expression.

Graph of Zero Polynomial

By looking at the places on the graph where the graph line intersects the x-axis, one can determine a polynomial's zeros.

Solved Examples

Example 1: What is the value of ‘a’ when the degree of the polynomial, $x^3+x^{a-4}+x^2+1$, is $4$?

Solution: The highest power of $x$ in a polynomial $P(x)$ is called the degree $(x)$.

therefore , $x^{a-4}=x^4$

$a-4=4,a=4+4=8$

Hence, the value of a comes out to be $8$.

Example 2: Sam is aware that a quadratic polynomial has zeros of -3 and 5. How can we assist in deriving the polynomial equation?

Solution: The given zeros of the quadratic polynomial are $-3$ and $5$.

Consider $\alpha =-3$, and $\beta =5$

Then, calculate the sum of the roots $=\alpha +\beta =2$

Product of the roots $=\alpha .\beta =-15$

Since, the required quadratic equation is $x^2-(\alpha +\beta )x+\alpha .\beta =0$

Put the values of the zeros in the equation above

$-2(x)+(-15)=0$

Hence, $x^2-2x-15=0$ is the required equation.

Practice Questions

1. Find the polynomial with the values -2 and -3 for the zeros.

1. $x^2-5x-6=0$

2. $x^2+6x+5=0$

3. $x^2+5x+6=0$

4. $x^2-5x+6=0$

Answer: C

2. A polynomial's zeros are also known as the equation's____

1. Variables

2. Roots

3. Constants

4. Exponents

Answer: D

Summary

Let's review what we learnt from this article. All x-values that reduce a polynomial, p(x), to zero are considered zeros. They are intriguing to us for a variety of reasons, one of which is because they show the graph's x-intercepts for the polynomial. Their relationship to the polynomial factors is direct. This article discussed the geometric meaning of a polynomial's zeros and how to find them. For you to better comprehend this idea, we have included practice problems and examples with answers that have been solved.