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104 LearnersLast updated on September 26, 2025
Linear, Quadratic, and Cubic Polynomials
Polynomials are expressions consisting of variables, constants, and exponents. Based on their degrees, polynomials are classified into different types. In this article, we will learn about the three main types of polynomials.
What are Linear, Quadratic, and Cubic Polynomials?
Linear, quadratic, and cubic are the three main types of polynomials. We shall learn more about them in the following sections.
What are Linear Polynomials?
A polynomial with a degree of one is known as a linear polynomial. Here, the highest exponent of the expression is one. It is of the form p(x) = ax + b, where a ≠ 0. Examples of linear polynomials are:
p(x) = 3x + 2
q(x) = 𝜋y + 5
r(z) = -8z
What are Quadratic Polynomials?
A polynomial where the highest exponent is 2 is known as a quadratic polynomial. The quadratic polynomial will be in the form of p(x) = ax2 + bx + c, where a ≠ 0. Examples of quadratic polynomials are:
p(x) = 5x2 + 2x + 2
q(y) = y2 - 2
s(z) = 7z2 + z
What are Cubic Polynomials?
In a cubic polynomial, the highest exponent of the variable is 3. Its general form is p(x) = ax3 + bx2 + cx + d, where a ≠ 0. Here are a few examples of cubic polynomials:
- p(x) = x3 + 4x2 - 3x + 2
- q(x) = 2x3 - 6
- r(x) = x3 + 2x
Real Life Applications of Linear, Quadratic, and Cubic Polynomial
Polynomials are widely used in real-life situations. From calculating how much something costs to predicting how a ball moves, here are some of the examples of the real-life applications of linear, quadratic, and cubic polynomials.
- Mathematics & Algebra: Linear polynomials are used in solving equations, such as calculating cost or solving word problems. Quadratic polynomials are used to solve problems like finding the area of a rectangle when only the relationship between the length and the width is known. Also, cubic polynomials can appear when finding how much water a box-shaped tank can hold, especially if the dimensions depend on a variable.
- Engineering & Construction: In engineering and construction, linear polynomials are used to estimate materials, quadratic equations are used in designing curved structures like arches and bridges, and cubic polynomials help design complex shapes like road curves and ramps.
- Technology & Computer Graphics: Linear polynomials can be used to create simple animations. For example, quadratic polynomials are used to draw smooth curves, and cubic polynomials create realistic 3D movements.
Common Mistakes and How to Avoid Them in Linear, Quadratic, and Cubic Polynomials
When learning about polynomials, especially linear, quadratic, and cubic types, it is normal to make mistakes. That’s why it’s important to learn about a few common mistakes beforehand, so that we can avoid them in the future.
Mistake 1
Forgetting to move terms
Sometimes students might forget to shift the terms from LHS to RHS or vice versa. Always move all the variable terms to one side and constants to the other to solve the equation.
Mistake 2
Mistakes in signs
Not changing the signs when moving to the other side is a mistake. Remember that moving the term from one side of the equation to the other results in a change of signs. For example, +3 from the left side will become -3 when it moves to the right side.
Mistake 3
Not factoring correctly
Factoring x2 + 5x + 6 as (x + 2)(x + 4) instead of (x + 2)(x + 3) is wrong. Cross-check by multiplying the factors to see if it gives the original equation. If not, then the factors are incorrect.
Mistake 4
Forgetting all possible solutions
Finding only one solution and concluding it as a final answer leads to a mistake. Always set the factors equal to 0 and find a solution for each factor.
Mistake 5
Thinking every cubic polynomial has three roots
Expecting 3 roots from a quadratic equation, where only one is a real number and the others are imaginary. The cubic polynomial can have one, two, or three real solutions.
Solved Examples of Linear, Quadratic, and Cubic Polynomials
Problem 1
Solve the linear equation: 2x + 5 = 11
x = 3
Explanation
To find the value of x, the first step is to subtract 5 on both sides of the equation.
2x + 5 - 5 = 11 - 5
Now, the equation becomes:
2x = 6
Divide both sides by 2 to isolate x.
2x/2 = 6/2
x = 3
Problem 2
Solve the quadratic equation: x2 - 5x + 6 = 0
x = 2 or x = 3
Explanation
Let’s use the factorization method to solve the quadratic equation:
x2 - 5x + 6 = (x - 2)(x - 3)
Now set each factor to 0.
x - 2 = 0; x - 3 = 0
Therefore, x = 2, x = 3.
Problem 3
Solve the cubic equation: x3 - 6x2 + 11x - 6 = 0
x = 1, 2, 3
Explanation
Factorizing the polynomial, we get:
x3 - 6x2 + 11x - 6 = (x - 1)(x - 2)(x - 3)
Setting each factor to 0,
x - 1 = 0
x - 2 = 0
x - 3 = 0
Therefore, x = 1, 2, and 3.
Problem 4
Solve the quadratic equation: x2 + 7x + 10 = 0
x = -2 or x = -5
Explanation
Let’s use the factorization method again.
x2 + 7x + 10 = (x + 2)(x + 5)
Setting the equation to 0, we get:
x + 2 = 0
x = -2
x + 5 = 0
x = -5
Problem 5
Solve the linear equation: 3x - 9 = 0
x = 3
Explanation
We should add 9 on both sides of the equation to isolate 3x.
3x - 9 + 9 = 0 + 9
3x = 9
Now, divide both sides by 3 to isolate x.
3x/3 = 9/3
x = 3
FAQs on Linear, Quadratic, and Cubic Polynomials
1.What does ‘degree’ mean in a polynomial?
2.How do linear, quadratic, and cubic polynomials look on a graph?
3.How many solutions can a linear, quadratic, and cubic polynomial have?
4.What is a coefficient in a polynomial?
5.What is a constant?
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