Polynomials Explained
Why Do We Study Polynomials?
Before diving into polynomials, you're already familiar with basic algebraic expressions—adding, subtracting, multiplying, and dividing them. You've even factored expressions using identities like (x + y)² = x² + 2xy + y².
A polynomial is simply a specific type of algebraic expression. Understanding them is crucial because they are the building blocks for solving many complex mathematical problems in higher mathematics.
Part 1: The Core Definition (The Golden Rule!)
What separates a polynomial from any other algebraic expression? It all comes down to the exponents!
Variables vs. Constants
In algebra, we use:
Variables (like x, y, z) — can take any real value
Constants (like a, b, c) — keep the same value throughout a problem
📐 Real-World Example: Consider a square with side length x units.
- Perimeter = 4x (here, 4 is constant, x is variable)
- Area = x² (as the side varies, the area varies)
What Makes a Polynomial?
✓ Golden Rule: An expression is a polynomial in one variable (like x) ONLY if the exponents of the variable are WHOLE NUMBERS (0, 1, 2, 3, ...).
Is It a Polynomial?
✓ YES, These ARE Polynomials:
• 2x (exponent: 1) ✓
• x² + 2x (exponents: 2, 1) ✓
• x³ - x² + 4x + 7 (exponents: 3, 2, 1, 0) ✓
• x² + 2x (exponents: 2, 1) ✓
• x³ - x² + 4x + 7 (exponents: 3, 2, 1, 0) ✓
✗ NO, These are NOT Polynomials:
• x + 1/x = x + x⁻¹ (exponent -1) ✗
• 3√x + 5 = 3x^(1/2) + 5 (exponent 1/2) ✗
• Why? Exponents must be whole numbers!
• 3√x + 5 = 3x^(1/2) + 5 (exponent 1/2) ✗
• Why? Exponents must be whole numbers!
Part 2: The Parts of a Polynomial
Every polynomial is made up of terms, and each term has a coefficient.
Example: -x³ + 4x² + 7x - 2
Terms: -x³, 4x², 7x, -2
Coefficients:
• Coefficient of x³ is -1
• Coefficient of x² is 4
• Coefficient of x is 7
• Constant term -2 is the coefficient of x⁰ (remember x⁰ = 1)
Coefficients:
• Coefficient of x³ is -1
• Coefficient of x² is 4
• Coefficient of x is 7
• Constant term -2 is the coefficient of x⁰ (remember x⁰ = 1)
Polynomials by Number of Terms
| Type | Definition | Examples |
|---|---|---|
| Monomial (Mono = One) |
One term only | 2x, 5x³, y, 7 |
| Binomial (Bi = Two) |
Two terms only | x + 1, x² - x, y⁹ + 1 |
| Trinomial (Tri = Three) |
Three terms only | x + x² + π, y⁴ + y + 5 |
Part 3: The Degree of a Polynomial (Most Important!)
📌 Definition: The degree of a polynomial is the highest power of the variable in the expression.
| Polynomial | Highest Power | Degree | Type |
|---|---|---|---|
| 7 (Constant) | 0 | 0 | Constant Polynomial |
| 4x + 5 | 1 | 1 | Linear Polynomial |
| 2x² + 5 | 2 | 2 | Quadratic Polynomial |
| 4x³ | 3 | 3 | Cubic Polynomial |
| x⁵ - x⁴ + 3 | 5 | 5 | Higher Degree |
Standard Forms by Degree
Linear Polynomial: ax + b (where a ≠ 0)
Can have at most 2 terms. Example: 3x + 2
Can have at most 2 terms. Example: 3x + 2
Quadratic Polynomial: ax² + bx + c (where a ≠ 0)
Can have at most 3 terms. Example: 2x² + 5x + 3
Can have at most 3 terms. Example: 2x² + 5x + 3
Cubic Polynomial: ax³ + bx² + cx + d (where a ≠ 0)
Can have at most 4 terms. Example: x³ + 2x² - 3x + 1
Can have at most 4 terms. Example: x³ + 2x² - 3x + 1
Part 4: Zeroes and Roots of Polynomials
💡 Definition: A zero of a polynomial is a number that, when substituted for the variable, makes the polynomial equal to zero.
Finding Zeroes with Examples
Example 1: Find the value at x = 1
If p(x) = 5x² - 3x + 7
Then p(1) = 5(1)² - 3(1) + 7 = 5 - 3 + 7 = 9
If p(x) = 5x² - 3x + 7
Then p(1) = 5(1)² - 3(1) + 7 = 5 - 3 + 7 = 9
Example 2: Find a zero
Consider p(x) = x - 1
If we substitute x = 1: p(1) = 1 - 1 = 0
✓ Therefore, 1 is a zero of polynomial x - 1
Consider p(x) = x - 1
If we substitute x = 1: p(1) = 1 - 1 = 0
✓ Therefore, 1 is a zero of polynomial x - 1
Finding Zero of Linear Polynomials
For p(x) = ax + b (where a ≠ 0):
Set ax + b = 0
x = -b/a ← This is the zero!
Set ax + b = 0
x = -b/a ← This is the zero!
Example: Find the zero of p(x) = 2x + 1
Set 2x + 1 = 0
2x = -1
x = -1/2 ← The zero
Key Fact: A linear polynomial has exactly ONE zero!
Set 2x + 1 = 0
2x = -1
x = -1/2 ← The zero
Key Fact: A linear polynomial has exactly ONE zero!
Important Facts About Zeroes
• A zero of a polynomial doesn't have to be 0
• 0 may or may not be a zero
• Non-zero constant polynomials (like 5) have NO zero
• A polynomial can have MORE than one zero
• 0 may or may not be a zero
• Non-zero constant polynomials (like 5) have NO zero
• A polynomial can have MORE than one zero
Part 5: The Factor Theorem (Advanced Tool!)
🔑 Factor Theorem:
(x - a) is a factor of polynomial p(x) if and only if p(a) = 0
In simple words: If substituting 'a' gives you 0, then (x - a) is definitely a factor!
(x - a) is a factor of polynomial p(x) if and only if p(a) = 0
In simple words: If substituting 'a' gives you 0, then (x - a) is definitely a factor!
Example: Is (x + 2) a factor of x³ + 3x² + 5x + 6?
The zero of (x + 2) is -2
Calculate p(-2):
p(-2) = (-2)³ + 3(-2)² + 5(-2) + 6
p(-2) = -8 + 12 - 10 + 6 = 0
✓ Since p(-2) = 0, YES, (x + 2) IS a factor!
The zero of (x + 2) is -2
Calculate p(-2):
p(-2) = (-2)³ + 3(-2)² + 5(-2) + 6
p(-2) = -8 + 12 - 10 + 6 = 0
✓ Since p(-2) = 0, YES, (x + 2) IS a factor!
Part 6: Essential Algebraic Identities
These formulas help you factor and expand polynomials quickly without long multiplication.
| Identity Name | Formula | Use Case |
|---|---|---|
| Perfect Square (Sum) | (x + y)² = x² + 2xy + y² | Factoring trinomials |
| Difference of Squares | x² - y² = (x + y)(x - y) | Quick factorization |
| Trinomial Square | (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx | Three-term expansion |
| Cube Sum | (x + y)³ = x³ + y³ + 3xy(x + y) | Cubic expressions |
Quick Example: Factorize 49a² + 70ab + 25b²
Observe: 49a² = (7a)², 25b² = (5b)², 70ab = 2(7a)(5b)
Using Identity (x + y)²:
= (7a + 5b)² ✓
Observe: 49a² = (7a)², 25b² = (5b)², 70ab = 2(7a)(5b)
Using Identity (x + y)²:
= (7a + 5b)² ✓
🍰 Think of a Polynomial Like a Mathematical Recipe
Variables are like ingredients (flour, sugar) whose quantity can vary
Coefficients are the specific measurements (2 cups flour, 3 eggs)
Exponents are the cooking instructions (bake it squared or cubed)
Degree is the overall complexity (quick snack vs. elaborate dish)
If the recipe asks to bake it to power -1 or 1/2, it's no longer a polynomial recipe! 🚫
Coefficients are the specific measurements (2 cups flour, 3 eggs)
Exponents are the cooking instructions (bake it squared or cubed)
Degree is the overall complexity (quick snack vs. elaborate dish)
If the recipe asks to bake it to power -1 or 1/2, it's no longer a polynomial recipe! 🚫
Key Takeaways ⭐
1 A polynomial must have whole number exponents
2 Degree = highest power of the variable
3 Classify by terms: monomial, binomial, trinomial
4 Classify by degree: linear, quadratic, cubic
5 A zero makes the polynomial equal to 0
6 Use Factor Theorem for factorization
7 Algebraic identities save time in calculations
2 Degree = highest power of the variable
3 Classify by terms: monomial, binomial, trinomial
4 Classify by degree: linear, quadratic, cubic
5 A zero makes the polynomial equal to 0
6 Use Factor Theorem for factorization
7 Algebraic identities save time in calculations
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