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Zero Polynomial

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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


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.

  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.

Last updated date: 29th Sep 2023
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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.