![copy icon](/cdn/images/seo-templates/copy.png)
What is 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 $ax^2 + bx + c$, 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 $ax^3+ bx^2+ cx+ d$, 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](https://www.vedantu.com/seo/content-images/85bc420d-0ab3-4f29-b525-b8326ce79165.png)
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 $= α + \beta = 2$
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.
$x^2-5x-6=0$
$x^2+6x+5=0$
$x^2+5x+6=0$
$x^2-5x+6=0$
Answer: C
2. A polynomial's zeros are also known as the equation's____
Variables
Roots
Constants
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.
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
FAQs on Zero Polynomial
1. What connection exists between a quadratic polynomial's coefficients and its sum of zeros?
The negative of the coefficient of $x$ by the coefficient of $x^2$ is equal to the sum of the zeroes in a quadratic polynomial. The constant term divided by the coefficient of $x^2$ is equal to the product of the zeroes. The sum and Product of Zeros of a quadratic equation are as follows:
Sum of zeroes, $\alpha+\beta$ is $\dfrac{-b}{a}$
Product of zeroes, $\alpha \beta$ is $\dfrac{c}{a}$.
2. How can we find the complex zeros of a polynomial function?
The complex number formula, $i^2 = -1$, can be used to compute the complex zeros of polynomials. Using the value of I from complex numbers, the negative roots can also be made simpler. Finding the square root of a negative number is impossible for the equation ${(x + 3)}^2$ = -25. Here, we write ${(x + 3)}^2$ = 25$i^2$ using the formula $i^2$ = -1. After simplifying, we get (x + 3) = + 5i, and the polynomial's zeros are -3 + 5i and -3 -5i.
3. Do all polynomials include a zero?
There could be zero, one, or several zeros in a polynomial function. All positive, odd-order polynomial functions have at least one zero, whereas positive, even-order polynomial functions might not. Any polynomial of positive order, regardless of odd or even, can have a maximum number of zeros equal to its order.