Polynomials and Rational Expressions

TL;DR

Polynomials are expressions with variables raised to non-negative integer powers, combined with addition, subtraction, and multiplication. Rational expressions are fractions where the numerator and denominator are polynomials. You'll learn to simplify, add, subtract, multiply, and divide both types of expressions.

1. The Mental Model

Think of polynomials as building blocks, like single words. Rational expressions are like sentences made from these words. You're learning the grammar rules to combine, break apart, and understand these mathematical "sentences."

2. The Core Material

What's a Polynomial?

Photo by Vitaly Gariev on Pexels

A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Examples:
* 3x + 2
* x^2 - 5x + 6
* 7y^3 + 2y - 1

Key terms:
* Term: Each part of the polynomial separated by a plus or minus sign (e.g., 3x, 2).
* Coefficient: The number multiplying a variable (e.g., 3 in 3x).
* Degree: The highest exponent of the variable in a polynomial (e.g., 2 in x^2 - 5x + 6).

Operations with Polynomials

Photo by Vitaly Gariev on Pexels

Adding and Subtracting Polynomials

Combine like terms. Like terms have the exact same variable parts (same variable, same exponent).

Example:
(3x^2 + 2x - 1) + (x^2 - 5x + 4)
= 3x^2 + x^2 + 2x - 5x - 1 + 4
= 4x^2 - 3x + 3

Multiplying Polynomials

Use the distributive property (FOIL for two binomials, or distribute each term of the first polynomial to every term of the second).

Example (Binomial by Binomial - FOIL):
(x + 2)(x - 3)
= x*x + x*(-3) + 2*x + 2*(-3)
= x^2 - 3x + 2x - 6
= x^2 - x - 6

Example (Polynomial by Polynomial):
(x + 1)(x^2 - 2x + 3)
= x(x^2 - 2x + 3) + 1(x^2 - 2x + 3)
= x^3 - 2x^2 + 3x + x^2 - 2x + 3
= x^3 - x^2 + x + 3

Dividing Polynomials (Long Division & Synthetic Division)

• Long Division: Similar to long division with numbers. Useful for dividing by any polynomial.
• Synthetic Division: A shortcut for dividing by a linear binomial of the form (x - k).

What's a Rational Expression?

Photo by Ann H on Pexels

A rational expression is a fraction where both the numerator and the denominator are polynomials.

Examples:
* (x + 1) / (x - 2)
* (x^2 - 4) / (x + 2)

Important: The denominator cannot be zero. You'll often need to find values that make the denominator zero and exclude them from the domain.

Operations with Rational Expressions

Photo by Vanessa Garcia on Pexels

Simplifying Rational Expressions

Factor the numerator and denominator, then cancel out common factors.

Example:
(x^2 - 4) / (x + 2)
= ((x - 2)(x + 2)) / (x + 2)
= x - 2, provided x != -2

Multiplying Rational Expressions

Multiply the numerators together and the denominators together. Simplify before or after.

Example:
(x + 1)/x * (x^2)/(x + 1)
= (x + 1)x^2 / x(x + 1)
= x, provided x != 0, x != -1

Dividing Rational Expressions

Multiply by the reciprocal of the second expression (flip the second fraction).

Example:
(x - 3)/2 / (x - 3)/4
= (x - 3)/2 * 4/(x - 3)
= (4(x - 3)) / (2(x - 3))
= 4/2
= 2, provided x != 3

Adding and Subtracting Rational Expressions

Find a common denominator, then add/subtract the numerators.

Example:
1/x + 2/(x + 1)
Common denominator is x(x + 1).
= (1(x + 1))/(x(x + 1)) + (2x)/(x(x + 1))
= (x + 1 + 2x) / (x(x + 1))
= (3x + 1) / (x(x + 1))

graph TD
    A["Start: Polynomial or Rational Expression?"] --> B{Is it a Polynomial?}
    B -- Yes --> C["Polynomial Operations"]
    C --> C1["Add/Subtract: Combine Like Terms"]
    C --> C2["Multiply: Distribute (FOIL)"]
    C --> C3["Divide: Long or Synthetic Division"]
    B -- No --> D["Rational Expression Operations"]
    D --> D1["Simplify: Factor & Cancel"]
    D --> D2["Multiply: Multiply Numerators/Denominators"]
    D --> D3["Divide: Multiply by Reciprocal"]
    D --> D4["Add/Subtract: Find Common Denominator"]
    C1 --> E["End"]
    C2 --> E
    C3 --> E
    D1 --> E
    D2 --> E
    D3 --> E
    D4 --> E

3. Worked Example

Let's simplify and combine two rational expressions:
((x^2 - 1) / (x^2 + 2x + 1)) * ( (x + 1) / (x - 1) ) + 3/x

First, simplify the multiplication part:
((x - 1)(x + 1) / (x + 1)(x + 1)) * ( (x + 1) / (x - 1) )

Cancel common factors in the first fraction:
(x - 1) / (x + 1)

Now, perform the multiplication:
((x - 1) / (x + 1)) * ( (x + 1) / (x - 1) )
Cancel (x - 1) and (x + 1):
1

So the expression simplifies to 1 + 3/x.

Now, add these two terms by finding a common denominator:
1 + 3/x
= x/x + 3/x
= (x + 3)/x

This is valid for x != 1, x != -1, and x != 0.

4. Key Takeaways

• Polynomials involve non-negative integer exponents and basic arithmetic (addition, subtraction, multiplication).
• Rational expressions are fractions of polynomials, with the denominator not equal to zero.
• When adding or subtracting polynomials, only combine terms with the exact same variable part.
• Factoring is crucial for simplifying rational expressions and for multiplying/dividing them efficiently.
• To add or subtract rational expressions, you must find a common denominator first.
• Dividing rational expressions means multiplying by the reciprocal of the divisor.
• Always note the restrictions on the variable for rational expressions, where the denominator would be zero.

Common Mistakes to Avoid:
- Trying to add or subtract unlike terms in polynomials.
- Forgetting to distribute negative signs when subtracting polynomials or rational expressions.
- Canceling terms that are added or subtracted, instead of only factors (e.g., (x+1)/x is NOT 1+1/x).
- Not flipping the second fraction when dividing rational expressions.

5. Now Try It

Simplify the following expression completely:
((x^2 - 9) / (x^2 + 6x + 9)) / ((x - 3) / (2x + 6))

What to do:
1. Factor all numerators and denominators in both rational expressions.
2. Rewrite the division as multiplication by the reciprocal.
3. Cancel out any common factors in the numerator and denominator.
4. Write down any restrictions on x.

What success looks like: A single, simplified rational expression and a list of x values that the original expression cannot be equal to.