** Graphs of polynomials** provide a visual representation of polynomial functions. The graphs of polynomials play a vital role in some applications like science, finance engineering, etc.

In this article, we will cover what a polynomial is, what is a graph of a polynomial, what are the types of polynomial functions, how to make a graph of different types of polynomials, what are real-life uses of the graph of a polynomial function and conclusion of the polynomial.

Table of Content

**What is a Polynomial?**

**What is a Polynomial?**

A polynomial is an algebraic expression in which the power of a variable is a non-negative integer. A polynomial is represented as p(x) which means that the polynomial expression is in terms of variable x.

**For Example,**

- p(x) = 5x – 2 is a polynomial in a variable x.
- t(y) = 6x
^{2 }+ 8y – 4 is polynomial in variable y.

**Learn, ****Polynomial**

**Graph of a Polynomial**

**Graph of a Polynomial**

A polynomial of graphs is shown on x y coordinate plans. We can represent the polynomial in the form of a graph. In graphs of a polynomial, we should know how to draw different types of polynomials on a graph and what real uses of graphs are in a polynomial.Â

**Types of Polynomial Functions**

**Types of Polynomial Functions**

Polynomials are fundamentals in algebra expression. The different types of polynomial functions:

- Constant Function
- Linear Function
- Quadratic Function
- Cubic Function
- Bi-Quadratic Function

**Check ****Types of Polynomials**

** 1. Constant Polynomial-** A polynomial whose degree is zero is called a constant polynomial.

** For examples**:

- g(x) = 5
- p(y) = -8
- n(t) = 5/9

** 2. Linear Polynomial**–Â A polynomial whose degree is 1 is called a linear polynomial.

** For examples**:

- j(x) = 6x + 2
- p(y) = 2y
- h(z) = 9z

** 3. Quadratic Polynomial-** A polynomial whose degree is 2 is called a quadratic polynomial.

** For examples**:

- f(x) = 4x
^{2}+ 5 - g(y) = 8/5y
^{2}– 3y

** 4. Cubic Polynomial**– A polynomial whose degree is 3 is called a cubic polynomial.

** For examples:**Â

- p(x) = 4x
^{3}+ 5x – 8 - h(y) = 9/5y
^{3}+ 2y^{2}– 7

** 5. Bi-Quadratic Polynomial**– A polynomial whose degree is 4 is called bi-quadratic polynomial.

**For examples:**

- p(x) = 4x
^{4}+ 8 - f(z) = 2z
^{4}– 6z^{2}+ 7

**How to Draw a Graph of a Polynomial?**

**How to Draw a Graph of a Polynomial?**

Drawing the graph of a polynomial involves several steps.

** Step 1: **Know the form of the polynomial, , where ( n ) is the degree of the polynomial.

** Step 2: **Determine the degree of the polynomial to understand the overall shape and behavior of the graph. Note the leading coefficient (a

_{n}).

** Step 3: **Calculate and mark the x-intercepts by setting f(x) = 0 and solving for ( x ). Also, find the y-intercept by setting (x = 0).

** Step 4:** Identify the end behavior by looking at the degree and leading coefficient. For even-degree polynomials, the ends go in the same direction; for odd-degree polynomials, they go in opposite directions.

** Step 5: **Determine turning points (where the graph changes direction) by finding the critical points where f'(x) = 0 or is undefined. Use these points to sketch the curve.

** Step 6: **Even-degree polynomials may exhibit symmetry about the y-axis, while odd-degree polynomials may show symmetry about the origin.

** Step 7: **Plot the identified points, including intercepts, turning points, and any additional points of interest. Connect the points smoothly to sketch the graph.

### Graph of Constant Polynomial

The graph of a constant polynomial is a horizontal line parallel to the x-axis. A constant polynomial has the form f(x) = c, where (c) is a constant. The graph represents a straight line that does not slope upward or downward; it remains at a constant height across all values of (x).

The graph is a horizontal line at the height corresponding to the constant term (c).**Horizontal Line:**

: Since the function is a constant, there is no change in the y-values as (x) varies. The line is perfectly level.**No Slope**

Â Unless the constant term is zero (c = 0), there are no x-intercepts, and the line intersects the y-axis at the constant value (c).**No Intercepts:**

** Â For Example:** y = 2

### Graph of Linear Polynomial

The graph of a linear polynomial, which is a polynomial of degree 1, has the following features:

: The graph is a straight line.**Straight Line**

: It has exactly one root or x-intercept.**One Root/Zero**

: The slope of the line remains constant.**Constant Slope**

** For example**: y = 2x + 5, a = 2 and b = 5

### Graph of Quadratic Polynomial

The graph of a quadratic polynomial, which is a polynomial of degree 2, has some features:

: The parabola is symmetric with respect to its axis of symmetry.**Symmetry**

: The quadratic polynomial may have two x-intercepts, one x-intercept or no x-intercepts.**Intercepts**

: The graph is a parabola, which can either be open upwards or downwards.**Parabolic Shape**

** For example**, y = 3x

^{2}+ 2x – 7

### Graph of Cubic Polynomial Function

The graph of a cubic polynomial, which is a polynomial of degree 3, has some features:

The graph will exhibit an “S” shape.**Cubic Shape:**

It may have up to two turning points.**Turning Points:**

It can have up to three real roots and intercepts with the x-axis.**Intercepts:**

** For Example,** p(x)=x

^{3}âˆ’3x

^{2}âˆ’4x+12

**How to Find Roots using Graph of Polynomial Function**

**How to Find Roots using Graph of Polynomial Function**

Finding the roots (or zeros) of a polynomial function from its graph involves identifying the x-values where the graph intersects the x-axis. The roots are the values of * x* for which the function equals zero. Here’s a step-by-step guide:

** Step 1: **Start with the given polynomial function in standard form. For example, (ax

^{2}+ bx + c).

** Step 2: **Identify the coefficients (a), (b), and (c) in the polynomial. These coefficients are crucial for using the quadratic formula.

** Step 3: **Apply the quadratic formula:

** Step 4: **Evaluate the discriminant (b

^{2}– 4ac). The discriminant determines the nature of the roots:

- If (Î” > 0), there are two distinct real roots.
- If (Î” = 0), there is one real root (a repeated root).
- If (Î” < 0), there are two complex (conjugate) roots.

** Step 5: **Simplify the square root part of the formula. If the discriminant is positive, take the square root. If it’s negative, express it in terms of (i), the imaginary unit.

** Step 6: **Use the âˆ“ symbol to represent both the positive and negative square root solutions.

** Step 7: **Plug in the values of (a), (b), and (c) into the quadratic formula and perform the calculations.

** For Example, **p(x)=2x

^{2}âˆ’5x+2Â

To find the roots of the polynomial function p(x) = 2x^{2 }– 5x + 2 , use the quadratic formula. The quadratic formula is given by:

In the equation (ax^{2} + bx + c = 0), the coefficients are:

a = 2, b = -5, c = 2

put these values of a, b, and c in the formula,

This gives two solutions:

- For the positive square root: x = (5+3)/4 = 2
- For the negative square root: x = (5-3)/4 = 1/2

So, the roots of the polynomial function p(x) = (2x^{2} – 5x + 2) are (x = 2) and (x = 0.5)

**Learn, ****Roots of a Polynomial**

**Real-Life Uses of Graph of the Polynomial**

**Real-Life Uses of Graph of the Polynomial**

Some real-life uses of graph of polynomial are:

Businesses use polynomial graphs to analyze profit functions, helping them understand how changes in factors like pricing and production affect overall profit.**Profit Analysis:**

Governments and organizations use polynomial models to create budget plans. The graphs show the financial impact of different variables, aiding in effective resource allocation.**Budget Planning:**

Engineers use polynomial graphs to model and optimize designs. This is crucial in fields like structural engineering to ensure stability and efficiency.**Engineering Designs:**

Economists use polynomial functions to model economic trends. Graphs help predict changes in factors like inflation and employment over time.**Economic Trends:**

In medical research, polynomial graphs assist in modeling the growth of diseases or the effectiveness of treatments, providing valuable insights for healthcare planning.**Medical Research:**

Environmental scientists use polynomial functions to model and predict ecological changes. This aids in understanding the impact of human activities on the environment.**Environmental Studies:**

Physicists use polynomial equations to model physical phenomena. The resulting graphs help visualize and analyze experimental data, enhancing our understanding of the natural world.**Physics Experiments:**

Demographers use polynomial models to study population growth. Graphs assist in predicting population changes and planning for future needs.**Population Studies:**

Investors and financial analysts use polynomial graphs to analyze stock market trends. This helps in making informed investment decisions based on historical data.**Stock Market Analysis:**

Polynomial models are applied in criminal justice to analyze crime rates over time. This information is vital for planning law enforcement strategies and resource allocation.**Criminal Justice Planning:**

**Also, Check**

**Examples on Graph of Polynomial**

**Examples on Graph of Polynomial**

**Example 1. Find the value of a, if x â€“ a is a factor of x**^{3}** â€“ ax**^{2}** + 5x + a â€“ 3.**

**Solution:**

Let p(x) = x

^{3}â€“ ax^{2}+ 5x + a â€“ 3Given that x â€“ a is a factor of p(x).

â‡’ p(a) = 0

Â a

^{3}â€“ a(a)^{2}+ 5a + a â€“ 3 = 0Â a

^{3}â€“ a^{3}+ 5a + a â€“ 3= 0( a

^{3}â€“ a^{3}= 0)6a â€“ 3 = 0

6a = 3, a = 2

Therefore, a = 2.

**Example 2. Graph the polynomial function.**

**f(x) = 5x**^{4}** – xÂ² + 3**

**Graph of Polynomial – Practice Questions**

**Graph of Polynomial – Practice Questions**

** Q1**.

**Solve the quadratic equation: x**

^{2}

**+ 2x – 4 = o for x.****Q2. A polynomial of degree n has:**

**only one zero****At least n zeroes****More than n zeroes****At most n zeroes**

**Q3. If the zeroes of the polynomial x**^{2}** + px + q are double in value to the zeroes of 5x**^{2}** – 6 – 4. Find the value of p and q.**

**Q4. Draw the graphs of the polynomial f(x) = x**^{3}** – 5.**

**Graph of Polynomial – FAQs**

**Graph of Polynomial – FAQs**

### 1. Define Polynomial.

A polynomial is a mathematical expression consisting of variables raised to non-negative integer powers, combined with coefficients and addition or subtraction operations.

**2. What are Types of Polynomials?**

**2. What are Types of Polynomials?**

Â The different types of polynomials are as follow:

- Constant PolynomialÂ
- Linear Polynomial
- Quadratic Polynomial
- Cubic Polynomial
- Bi-Quadratic polynomial

**3. What are Key Features in Graphs of Polynomials?**

**3. What are Key Features in Graphs of Polynomials?**

Some key features in graphs of polynomials are as follow:

- Turning Points- The position where the graph changes direction.
- Intercepts-Â Points where x-axis and y-axis intersect in a graph.
- Symmetry- Such as odd and even symmetry.

**4. How to Find Zero of a Polynomial Function?**

**4. How to Find Zero of a Polynomial Function?**

To find the zeros of a polynomial function:

Set the function to zero: f(x) = 0.Step 1:Factorize the polynomial if possible.Step 2:Use the Zero-Product Property: Set each factor equal to zero and solve for \( x \).Step 3:Solve for ( x ) to find the zeros.Step 4:Check for repeated roots.Step 5:Use the quadratic formula for quadratic polynomials (axStep 6:^{2}+ bx + c): (x = )

**5. What is Significance of x-Intercepts on a Polynomial Graph?**

**5. What is Significance of x-Intercepts on a Polynomial Graph?**

The x-intercepts represent the points where the polynomial function equals zero. In practical terms, these points indicate the real-life values where the phenomenon modeled by the polynomial has a specific impact or occurrence.

**6. How to identify Degree of a Polynomial by looking its Graph?**

**6. How to identify Degree of a Polynomial by looking its Graph?**

By examining the graph, you can count the number of times the graph intersects the x-axis to identify the degree, as each intercept corresponds to a factor of the polynomial.

### 7. What is Graph of a Quadratic Polynomial?

The graph of a quadratic polynomial, which has the form (ax

^{2}+ bx + c), is a parabola. Depending on the sign of the coefficient (a), the parabola can open upwards (a > 0) or downwards (a < 0).

### 8. What is Nature of Curve of a Cubic Polynomial?

The nature of the curve can include S-shaped bends, loops, or multiple turning points, and it extends in both the positive and negative directions.

### 9. Which Polynomial has Line graph?

The graph of a linear polynomial is characterized by a straight line with a constant slope, and it does not show the curvature seen in higher-degree polynomials.