How Can You Identify a Cubic Polynomial by Its Degree and Terms?
The concept of cubic polynomials is essential in mathematics and helps in solving real-world and exam-level problems efficiently.
Understanding Cubic Polynomials
A cubic polynomial refers to an algebraic expression where the highest power of the variable is 3. This means it has a degree of three and is written in the standard form as \( ax^3 + bx^2 + cx + d \), where \( a \neq 0 \). This concept is widely used in solving cubic equations, understanding polynomial roots, and factorising algebraic expressions.
Formula Used in Cubic Polynomials
The standard formula for a cubic polynomial is: \( ax^3 + bx^2 + cx + d \), where:
• d is a constant
• a ≠ 0
To find the roots (also called zeros) of a cubic polynomial, set the polynomial equal to zero: \( ax^3 + bx^2 + cx + d = 0 \).
Here’s a helpful table to understand cubic polynomials more clearly:
Cubic Polynomial Examples Table
| Polynomial | Degree | Cubic? |
|---|---|---|
| \( x^3 - 6x^2 + 11x - 6 \) | 3 | Yes |
| \( 2x + 1 \) | 1 | No |
| \( 5x^3 + x^2 - 3x + 7 \) | 3 | Yes |
| \( 7x^2 - 4x + 9 \) | 2 | No |
This table shows how only those expressions with the highest power 3 are called cubic polynomials.
Worked Example – Solving a Cubic Polynomial
Example: Solve the cubic equation \( x^3 - 2x^2 - 8x - 35 = 0 \) if \( (x - 5) \) is a factor.
1. Write the given cubic equation in standard form: \( x^3 - 2x^2 - 8x - 35 = 0 \ )2. Since \( (x - 5) \) is a factor, use synthetic division or factor theorem to divide the polynomial by \( (x-5) \).
3. The quotient found is \( x^2 + 3x + 7 \).
4. Factorise \( x^2 + 3x + 7 \) using the quadratic formula: \( x = \frac{-3 \pm \sqrt{9-28}}{2} = \frac{-3 \pm i\sqrt{19}}{2} \).
5. Thus, the roots of the equation are \( x = 5 \), \( x = \frac{-3 + i\sqrt{19}}{2} \), \( x = \frac{-3 - i\sqrt{19}}{2} \).
How to Factorise a Cubic Polynomial
To factorise cubic polynomials, follow these steps:
1. Find at least one real root using trial and error or Rational Root Theorem.2. Divide the cubic polynomial by the corresponding linear factor \( (x - r) \), where \( r \) is the real root.
3. You will get a quadratic quotient. Factorise it further using quadratic methods.
For more advanced factorisation, visit Factorisation of Algebraic Expressions.
How to Find Roots of Cubic Polynomials
The roots of a cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \) can be found as follows:
1. Guess possible rational roots using Rational Root Theorem.2. Test these roots by substituting into the polynomial.
3. Once a root is found, factor out \( (x - r) \).
4. Use quadratic formula to solve the remaining quadratic part.
Alternatively, you can use online tools such as Cubic Equation Solver for fast answers.
Graph of Cubic Polynomial
The graph of a cubic polynomial usually has one or two turning points, and can cross the x-axis up to three times. If the leading coefficient \( a \) is positive, the graph rises from left to right. If \( a \) is negative, it falls from left to right. For visual understanding, connect this concept with Graphs of Polynomials.
The number of x-intercepts corresponds to the real roots of the polynomial.
Practice Problems
- Which of the following are cubic polynomials? \( x^3 + 4x^2 + 5x - 2 \), \( x^2 + 7x + 1 \), \( 6x^3 - 11x + 4 \)
- Find all roots of the cubic polynomial \( y^3 – 2y^2 – y + 2 \).
- Factorise \( z^3 + 8z^2 + 17z + 10 \).
- If \( (x + 2) \) is a factor of \( x^3 + 7x^2 + 14x + 8 \), find the other roots.
Common Mistakes to Avoid
- Confusing the degree with the number of terms – degree 3 means the highest exponent is 3, not that there are three terms.
- Forgetting to check all possible rational roots before factorising.
- Omitting complex roots when using the quadratic formula.
Real-World Applications
Cubic polynomials appear in physics (projectile motion), engineering (calculation of volumes), computer graphics (Bezier curves), and economics (cost and revenue functions). Learning to solve cubic polynomials with Vedantu prepares students for these real-world contexts.
We explored the idea of cubic polynomials, how to identify and solve them, practice with examples, and connect them to real-life applications. Practice more with Vedantu to build confidence in these concepts.
Related Concepts and Further Learning
- Degree of Polynomial
- Polynomial Equations
- Zeroes of Polynomials
- Factorisation of Algebraic Expressions
- Algebraic Expressions and Identities
FAQs on What Is a Cubic Polynomial?
1. How do you know if a polynomial is cubic?
A polynomial is called cubic if the highest exponent (degree) of its variable is 3. This means the general form is ax3 + bx2 + cx + d, where 'a' is non-zero. Check whether the largest exponent on 'x' in the equation is 3 to identify a cubic polynomial.
2. Is 7x + 3 a cubic polynomial?
No, 7x + 3 is not a cubic polynomial as the highest power of x is 1, making it a linear polynomial. A cubic polynomial must have x raised to the power of 3.
3. Is a cubic polynomial always degree of 3?
Yes, a cubic polynomial is always of degree 3. The term 'cubic' comes from the word 'cube', referring to an exponent of 3. Any polynomial of degree 3, that is, with a term ax3, is cubic.
4. What are the characteristics of a cubic polynomial?
Cubic polynomials have the following characteristics:
• The highest power of the variable is 3.
• The general form is ax3 + bx2 + cx + d, where a ≠ 0.
• Their graphs have at most 2 turning points and can have up to 3 real roots.
• They can be factored by factorisation methods or solved for roots.
5. What is a cubic polynomial? Give an example.
A cubic polynomial is a polynomial of degree 3 and has the general form ax3 + bx2 + cx + d, where a ≠ 0. For example, 2x3 - 5x2 + 4x - 7 is a cubic polynomial.
6. How do you factorise a cubic polynomial?
To factorise a cubic polynomial, follow these steps:
1. Use the factor theorem to find one root.
2. Divide the cubic polynomial by (x - root) using synthetic or long division.
3. Factorise the resulting quadratic, if possible.
For example, for x3 - 6x2 + 11x - 6, find one factor, divide, and then factorise the quadratic.
7. What is the general formula for a cubic polynomial?
The general formula for a cubic polynomial is:
ax3 + bx2 + cx + d, where a, b, c, d are real numbers and a ≠ 0.
8. What is the cubic formula for roots of a cubic equation?
The cubic formula gives the roots of a cubic equation ax3 + bx2 + cx + d = 0. The formula is complex, involving radicals. For most school-level problems, use factorisation or the factor theorem to find roots. Advanced solutions use Cardano's formula.
9. How does the graph of a cubic polynomial look?
The graph of a cubic polynomial is an S-shaped curve which can have up to two turning points and cross the x-axis up to three times. The shape depends on the coefficient 'a': if a is positive, the ends of the graph go in opposite directions (left down, right up); if a is negative, the opposite is true.
10. What is the product of roots of a cubic equation?
The product of the roots of a cubic equation ax3 + bx2 + cx + d = 0 is -d/a, where a ≠ 0. This is derived from Vieta's formulas.
11. How many real roots can a cubic polynomial have?
A cubic polynomial can have either one real root and two complex roots, or three real roots (which may or may not be distinct). The number of real roots depends on the coefficients and discriminant of the cubic equation.
12. Are all cubic polynomials factorable?
Not all cubic polynomials can be factorised over the real numbers. Some have roots that are complex or irrational, making them irreducible by simple factorisation methods. However, every cubic can be factorised over the complex numbers.
