The polynomial Formula gives the standard form of polynomial expressions. It specifies the arrangement of algebraic expressions according to their increasing or decreasing power of variables.
The General Formula of a Polynomial:
f(x) = anxn + an−1xn−1 + ⋯ + a1x + a0
Where,
- an, an−1, …, a1, a0 are the coefficients,
- x is the variable,
- n is the degree of the polynomial (the highest power of x).
What is Polynomial?
A polynomial is an algebraic expression consisting of terms with non-negative integer exponents of the variable. It can be expressed as the sum of monomials, binomials, or more complex expressions.
The degree of a polynomial is determined by the highest power of the variable in the expression. For example, in the polynomial f(x) = 3x2 + 4x + 5, the highest power of x is 2, so the degree of the polynomial is 2.
Like and Unlike Terms:
- Like terms: Terms that have the same variable raised to the same power (e.g., 3x2 and 5x2).
- Unlike terms: Terms that have different variables or different powers (e.g., 3x2 and 4x).
Types of Polynomial
Different Types of Polynomials have been discussed in the table below :
| Type of Polynomial | Description | General Formula | Example |
|---|---|---|---|
| Monomial | Polynomials with one term | axn | x, y², 3y³, etc. |
| Binomial | Polynomials with two terms | axn+ bym | 2x + y², x + 3y³, etc. |
| Trinomial | Polynomials with three terms | axn + bym + czk | 2x + z + y², z - x + 3y³, etc. |
| Quadratic Polynomial | Second-degree polynomial (typically two or three terms) | ax2+ bx + c | 2x + y² + 3y³, etc. |
Read More: Types of Polynomials (Based on Terms and Degrees)
Polynomial Identities
Let's learn some of the algebraic identities of polynomials and their expansion.
| Some Important Polynomial Identities |
|---|
| (x + y)² = x² + 2xy + y² |
| (x – y)² = x² – 2xy + y² |
| x² – y² = (x + y)(x – y) |
| (x + y)³ = x³ + y³ + 3xy(x + y) |
| (x – y)³ = x³ – y³ – 3xy(x – y) |
| x³ + y³ = (x + y)(x² – xy + y²) |
| x³ – y³ = (x – y)(x² + xy + y²) |
| x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx) |
| (x + a)(x + b) = x² + (a + b)x + ab |
| (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx |
Applications of Polynomial Formula
Polynomial Formula has the following applications :
- They are used to define the equation of various forces, paths, and other concepts in detail.
- Polynomial equations are used to explain unknown quantities and their relation with other quantities in detail.
- Polynomial formulas are used to solve various complex mathematical equations.
- They are used to estimate the curves of the roller-coaster tracks to estimate the suitable curvature and height of the tracks.
- They are used to correctly estimate the stock markets and accordingly, shares can be purchased or sold.
Read More: Real-Life Applications of Polynomials
Related :
Solved Examples on Polynomial Formula
Example 1: Find the factors of the given polynomial x2 + 5x + 6
Solution:
Given polynomial,
x2 + 5x + 6
= x2 + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)So factors of given polynomial are (x + 2) and (x + 3)
Example 2: Find the factors of the given polynomial x2 + 3x - 4
Solution:
Given polynomial,
x2 + 3x - 4
= x2 + 4x - x - 4
= x(x + 4) - 1(x + 4)
= (x + 4)(x - 1)So factors of given polynomial are (x + 4) and (x - 1)
Example 3: Find the factors of the given polynomial x2 - 7x + 12
Solution:
Given polynomial,
x2 - 7x + 12
⇒ x2 - 4x - 3x + 12
⇒ x(x - 4) - 3(x - 4)
⇒ (x - 4)(x - 3)So factors of given polynomial are (x - 4) and (x - 3)
Example 4: Simplify (x2 + 6x + 9) / (x + 3)3
Solution:
Given, (x2 + 6x + 9) / (x + 3)3
Now simplifying,
x2 + 6x + 9
= x2 + 3x + 3x + 9
= x(x + 3) + 3(x + 3)
= (x + 3)(x + 3)
= (x + 3)2(x2 + 6x + 9) / (x + 3)3 = (x + 3)2 / (x + 3)
= 1/(x+3)
Example 5: Expand (3x - 11)3 using the cubic polynomial formula.
Solution:
We know that, (x – y)3= x3 – y3 – 3xy (x – y)
Now, (3x - 11)3
= (3x)3 - (11)3 - 3(3x)(11)(3x-11)
= 27x3 - 1331 - 9x(3x -11)This is the required expansion.
Example 6: Divide the polynomial x3 - 6x2 +3x + 10 by x + 1
Solution:
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Check: Practice Questions on PolynomialsPolynomials
Practice Problems on Polynomials
Problem 1: Evaluate the polynomial at given values: P(x) = 3x4 − 5x3 + 2x2 − x + 7
- P(1)
- P(−2)
- P(0)
Problem 2: Factor the polynomial completely: x3 − 6x2 + 11x − 6
Problem 3: Find the roots of the polynomial: 2x2 − 4x − 6 = 0
Problem 4: Simplify the polynomial expression: (3x2 + 2x − 5) + (2x2 − 3x + 4)
Problem 5: Multiply the polynomials: (2x − 3)(x2 + x + 4)