Introduction to Polynomials

What is a Polynomial?

A polynomial is an expression with variables and coefficients, using only addition, subtraction, and multiplication.

Examples:

• 3x + 5
• x² − 4x + 7
• 2x³ + 5x² − x + 9

NOT polynomials:

• 1/x (division by variable)
• √x (fractional exponent)
• x⁻² (negative exponent)

Parts of a Polynomial

Term: A single part separated by + or −

• 5x², −3x, 7 are all terms

Coefficient: The number part of a term

• In 5x², the coefficient is 5
• In −3x, the coefficient is −3

Variable: The letter (usually x)

Constant: A term with no variable

• In x² + 3x + 7, the constant is 7

Example: Identify Parts

Polynomial: 4x³ − 2x² + 5x − 8

Terms: 4x³, −2x², 5x, −8

Coefficients: 4, −2, 5

Constant: −8

Variables: All have x

Degree of a Polynomial

Degree: The highest exponent of the variable

For a term: The exponent on the variable For a polynomial: The highest degree of any term

Example: Find Degree

5x³: Degree 3

2x: Degree 1 (x = x¹)

7: Degree 0 (constant)

Polynomial 3x⁴ + 2x² − 5x + 1:

• Highest exponent is 4
• Degree: 4

Types of Polynomials by Degree

Types by Number of Terms

Monomial: 1 term

• 5x²
• −7
• 3x

Binomial: 2 terms

• x + 5
• 3x² − 2x
• 4x³ + 7

Trinomial: 3 terms

• x² + 5x + 6
• 2x² − 3x + 1
• x³ + 2x − 4

Polynomial: Any number of terms

Standard Form

Standard form: Terms written in descending order of degree

Highest degree first, constant last

Example: Write in Standard Form

Given: 5 + 2x² − 3x + x³

Rearrange: x³ + 2x² − 3x + 5

Answer: x³ + 2x² − 3x + 5

Evaluating Polynomials

To evaluate: Substitute a number for the variable and calculate

Example 1: Evaluate P(x) = 2x² + 3x − 5 when x = 3

Substitute:

P(3) = 2(3)² + 3(3) − 5
P(3) = 2(9) + 9 − 5
P(3) = 18 + 9 − 5
P(3) = 22

Answer: 22

Example 2: Evaluate when x = −2

P(x) = x³ − 4x + 7

Substitute:

P(−2) = (−2)³ − 4(−2) + 7
P(−2) = −8 + 8 + 7
P(−2) = 7

Answer: 7

Adding Polynomials

Combine like terms: Terms with the same variable and exponent

Method: Add coefficients of like terms

Example 1: Add Polynomials

(3x² + 5x − 2) + (2x² − 3x + 7)

Step 1: Remove parentheses

• 3x² + 5x − 2 + 2x² − 3x + 7

Step 2: Group like terms

• (3x² + 2x²) + (5x − 3x) + (−2 + 7)

Step 3: Combine

• 5x² + 2x + 5

Answer: 5x² + 2x + 5

Example 2: Vertical Addition

Add: (4x² + 3x − 1) + (2x² − x + 5)

Align like terms:

  4x² + 3x − 1
+ 2x² −  x + 5
_______________
  6x² + 2x + 4

Answer: 6x² + 2x + 4

Subtracting Polynomials

Key: Distribute the negative sign!

Then combine like terms

Example 1: Subtract Polynomials

(5x² + 3x − 4) − (2x² + x − 1)

Step 1: Distribute negative

• 5x² + 3x − 4 − 2x² − x + 1

Step 2: Group like terms

• (5x² − 2x²) + (3x − x) + (−4 + 1)

Step 3: Combine

• 3x² + 2x − 3

Answer: 3x² + 2x − 3

Example 2: Watch the Signs!

(x³ − 2x + 5) − (x³ + 3x − 1)

Distribute:

• x³ − 2x + 5 − x³ − 3x + 1

Combine:

• (x³ − x³) + (−2x − 3x) + (5 + 1)
• 0 − 5x + 6
• −5x + 6

Answer: −5x + 6

Adding/Subtracting Multiple Polynomials

Example: Three Polynomials

(2x² + 3x) + (x² − 5x + 1) − (3x² − 2x)

Distribute and combine:

2x² + 3x + x² − 5x + 1 − 3x² + 2x
(2x² + x² − 3x²) + (3x − 5x + 2x) + 1
0x² + 0x + 1
1

Answer: 1

Finding Missing Terms

If a degree is missing, coefficient is 0

Example: x³ + 5x − 2

Missing: x² term

Can write as: x³ + 0x² + 5x − 2

Helpful for:

• Vertical addition/subtraction
• Identifying degree

Real-World Applications

Physics: Projectile motion

• h(t) = −16t² + 32t + 48 (height over time)

Economics: Cost functions

• C(x) = 5x² + 100x + 500 (production costs)

Geometry: Area and volume

• Area of complex shapes
• Volume formulas

Engineering: Design calculations

• Stress, load distributions

Practice