Polynomial and Rational Fractions

Introduction

Before learning partial fractions, it is important to understand two basic concepts: polynomials and rational fractions. These concepts are used to simplify algebraic expressions and solve mathematical problems in pharmacy.

We will start with simple examples and gradually move towards more challenging ones.

1. What Is a Polynomial?

A polynomial is an algebraic expression made up of constants and variables raised to non-negative whole-number powers, combined using addition, subtraction and multiplication.

A variable is a symbol, such as x, whose value can change.

Example:

3x² + 2x + 5

This is a polynomial because:

• 3x² means 3 × x × x.

• 2x means 2 × x.

• 5 is a constant, meaning its value does not change.

The numbers 3 and 2 are coefficients of their respective variable terms, while 5 is the constant term.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable with a non-zero coefficient.

Examples:

• 7 — Degree 0

• 2x + 3 — Degree 1

• x² + 4x + 1 — Degree 2

• 3x³ − x + 5 — Degree 3

Important: An expression containing a variable in the denominator, such as 1/x, or a negative power, such as x⁻¹, is not a polynomial.

2. What Is a Rational Fraction?

A rational fraction is a fraction in which both the numerator and denominator are polynomials, and the denominator must not be zero.

General form:

P(x) / Q(x)

Here:

• P(x) is the numerator polynomial.

• Q(x) is the denominator polynomial.

• Q(x) must not equal zero.

Example 1:

(x + 2) / (x − 3)

Numerator: x + 2

Denominator: x − 3

Both are polynomials, so this is a rational fraction. However, x cannot equal 3 because that would make the denominator zero.

Example 2:

(2x² + 3x + 1) / (x² + 4)

This is also a rational fraction because both the numerator and denominator are polynomials.

Tip: Whenever you see an algebraic fraction, identify its numerator and denominator first. Then check whether the denominator can become zero.

3. Solved Examples

Example 1 — Basic Level

Question: Identify the numerator and denominator of:

(3x + 5) / (x + 2)

Step 1: The expression above the fraction bar is the numerator.

Numerator = 3x + 5

Step 2: The expression below the fraction bar is the denominator.

Denominator = x + 2

Step 3: Both expressions are polynomials, so the given expression is a rational fraction.

Answer: It is a rational fraction, provided x ≠ −2.

Example 2 — Medium Level

Question: Is the following expression a rational fraction?

(x² − 4) / (x² + 3x + 2)

Step 1: Identify the numerator and denominator.

Numerator = x² − 4

Denominator = x² + 3x + 2

Step 2: Both are polynomials because all powers of x are non-negative integers.

Step 3: Factor the denominator.

x² + 3x + 2 = (x + 1)(x + 2)

The denominator becomes zero when x = −1 or x = −2.

Answer: Yes, it is a rational fraction, provided x ≠ −1 and x ≠ −2.

Example 3 — Challenging Level

Question: Classify each expression as a polynomial, a rational fraction, or neither.

(a) 2x³ − 5x + 4

(b) (x² + 1) / (x − 4)

(c) 1 / √x

Solution:

(a) 2x³ − 5x + 4

All powers of x are non-negative integers. Therefore, it is a polynomial.

(b) (x² + 1) / (x − 4)

The numerator and denominator are both polynomials. Therefore, it is a rational fraction, provided x ≠ 4.

(c) 1 / √x

This can be written as 1 / x^(1/2). Since the exponent 1/2 is not a non-negative integer, the expression is not a polynomial. It is also not a rational fraction of polynomials.

Answers:

(a) Polynomial

(b) Rational fraction

(c) Neither a polynomial nor a rational fraction of polynomials

4. Simplifying Rational Fractions

A rational fraction can sometimes be simplified by factoring the numerator and denominator and cancelling their common factors.

Example: Simplify the following expression.

(x² − 9) / (x² + 3x)

Step 1: Factor the numerator.

x² − 9 = (x − 3)(x + 3)

This uses the difference of squares formula:

a² − b² = (a − b)(a + b)

Step 2: Factor the denominator.

x² + 3x = x(x + 3)

Step 3: Write the expression in factored form.

[(x − 3)(x + 3)] / [x(x + 3)]

Step 4: Cancel the common factor (x + 3).

Final answer:

(x − 3) / x

The original expression requires x ≠ 0 and x ≠ −3. These restrictions remain even after simplification.

Important: Only common factors can be cancelled, not individual terms.

For example, (x + 3) / x cannot be simplified by cancelling x because x + 3 is a sum, not a product containing x as a common factor.

5. Practice Questions

Try solving these questions yourself.

1. What is the degree of 5x³ + 2x² − 7?

2. Which of the following is a rational fraction?

   (a) √x + 1

   (b) (x² + 1) / (x − 2)

   (c) x⁻² + 3

3. For which value is (x + 1) / (x − 5) undefined?

4. Simplify (x² − 16) / (x + 4), where the original expression is defined.

Answers

1. Degree 3

2. Option (b)

3. x = 5

4. x − 4, with the restriction x ≠ −4

Key Takeaways

A polynomial contains constants and variables raised to non-negative integer powers. A rational fraction is the ratio of two polynomials with a non-zero denominator.

Before simplifying a rational fraction, identify its numerator and denominator, check the restrictions on the variable, and factor carefully.

These concepts form the foundation for the next topic: Proper and Improper Rational Fractions.