Linear Polynomial
A linear polynomial is a type of polynomial where the highest degree of the variable is 1. In other words, the highest exponent of the variable is 1. Polynomials are algebraic expressions where the variables have non-negative integer powers. The expression consists of one or more terms such as a variable, constant, and a variable with a non-zero coefficient. Linear polynomials are the simplest form of polynomials. Let us learn more about linear polynomials and solve a few examples.
| 1 | Definition of Linear Polynomial |
| 2 | Roots of Linear Polynomial |
| 3 | Linear Polynomial Functions |
| 4 | Zero of Linear Polynomial |
| 5 | FAQs on Linear Polynomial |
Definition of Linear Polynomial
A linear polynomial is defined as any polynomial expressed in the form of an equation of p(x) = ax + b, where a and b are real numbers and a ≠ 0. In a linear polynomial, the degree of the variable is equal to 1 i.e., the highest exponent of the variable is one. Linear polynomial in one variable can have at the most two terms. The constraint that a should not be equal to 0 is required because if a is 0, then this becomes a constant polynomial. A few examples of a linear polynomial are: p(x): 2x + 3, q(y): πy + √2.
A polynomial is classified into 3 different types of polynomials based on the degree of the polynomial i.e. power of the leading term or the highest power of the variable.
Roots of Linear Polynomial
Roots of polynomials are the method of finding the value of the unknown variable. Once we determine the roots, we can evaluate the value of the polynomial to zero. In the case of linear polynomials, the root is exactly one with a ≠ 0. The formula for the root of linear polynomial for the expression ax + b is x = -b/a. Let us look at an example.
Example: Find the root of p(x) = 4x + 5
Solution: According to the roots of polynomials, a is the root of a polynomial p(x), if P(a) = 0. Thus, in order to determine the roots of polynomial p(x), we have to find the value of x for which p(x) = 0.
4x + 5 = 0
x = - 5/4.
Therefore, -5/4 is the root of the linear polynomial 4x + 5.
Proof of Roots of Linear Polynomial
To prove the roots of the linear polynomial formula, let us consider the general form of a linear polynomial p(x) = ax + b, where a and b are real numbers with a ≠ 0. The root of a polynomial p is the value x satisfying p(x) = 0.
Hence, p(x) = 0
ax + b = 0
x = -b/a.
Hence, proved.
Linear Polynomial Functions
Linear polynomials functions are also known as first-degree polynomials and are represented as y = ax + b. By definition, polynomial functions are expressions that might contain variables of differing degrees, non-zero coefficients, positive exponents, and constants. We can represent all the polynomial functions in the form of a graph. The below-given image shows the graphs of different polynomial functions. An important skill in coördinate geometry consists in recognizing the relationship between equations and their graphs.
- The graph of a linear polynomial function always forms a straight line and is represented as y = ax +b.
- The graph of a second-degree or quadratic polynomial function is a curve known as a parabola. It can be represented as y = ax2 + bx + c.
- A cubic polynomial function of the third degree has the form shown on the right and it can be represented as y = ax3 + bx2 + cx + d.
Solving Linear Polynomial Functions
To solving a linear polynomial function we need to equate the expression to 0 and solve for x as the main aim is to find the value of x. Hence, for any given function, p(y), its zeros are found by setting the function p(y) to zero. The values of y that represent the set equation are the zeroes of the function p(y). To find the zeros of a function, find the values of y where p(y) = 0.
For example: Consider the polynomial function f(y). Let us solve this function by first putting all the terms on one side and 0 on the other as shown below.
16y - 4 = 0
Simplifying it further, we get:
4y -1 = 0 ⇒ 4y = 1
Thus, y = 1/4 = [-2 ± 2√6]/2.
Zero of Linear Polynomial
As we know, a linear polynomial is of the following form p(x) = ax + b, where a ≠ 0. There is only one zero of this polynomial, and it is easy to find out that zero. We simply equate this polynomial to 0 and find out the corresponding value of x:
ax + b = 0
x = -b/a.
Note that a must not be equal to 0, otherwise this zero will have an undefined value (in fact, if a is equal to 0, then the original polynomial will not even be linear). Here are some examples: πx−3 = x = 3/π, √2x + 4 = x = - 4/√2 etc.
Related Topics
Listed below are a few topics related to linear polynomials, take a look.
FAQs on Linear Polynomials
How Do You Know if a Polynomial is Linear?
A polynomial of degree one is called the linear polynomial. That is, the highest exponent of the variable is one, then the polynomial is said to be a linear polynomial.
How to Classify Linear, Quadratic, and Cubic Polynomials?
Linear, quadratic and cubic polynomials can be classified on the basis of their degrees.
- A polynomial of degree one is a linear polynomial. For example, 5x + 3
- A polynomial of degree two is a quadratic polynomial. For example, 2x2 + x + 5
- A polynomial of degree three is a cubic polynomial. For example, y3 − 6y2 + 11y − 6
What is the Zero of a Linear Polynomial?
In order to find the zero of a linear polynomial equate P(x)= 0. Therefore the zero of a linear polynomial ax+b is -b/a.
What is the Formula to Find the Root of Linear Polynomial?
The formula for the root of linear polynomial for the expression ax + b is x = -b/a. For example, p(x) = 3x + 7, 3x + 7 = 0, x = - 7/3.
What is Linear Polynomial with Giving Example?
A linear polynomial is defined as any polynomial expressed in the form of an equation of p(x) = ax + b, where a and b are real numbers and a ≠ 0. For example: x -3, -12x + 5.