Unit 5 Polynomial Functions Homework 1 Monomials And Polynomials

Unit 5: Polynomial Functions - Homework 1: Monomials and Polynomials

This comprehensive guide dives deep into the world of monomials and polynomials, forming the foundation of Unit 5 on Polynomial Functions. We'll cover key concepts, provide examples, and offer strategies to master this crucial area of algebra. Understanding monomials and polynomials is essential for tackling more complex polynomial operations and applications later in the unit.

What are Monomials?

A monomial is a single term, which is a product of a constant and variables raised to non-negative integer powers. Let's break that down:

• Constant: A number (e.g., 5, -2, 1/3).
• Variables: Letters representing unknown values (e.g., x, y, z).
• Non-negative integer powers: Exponents that are whole numbers (0, 1, 2, 3...).

Examples of Monomials:

• 5x²
• -3y
• 7
• x³y²z
• ½ab

Examples that are NOT Monomials:

• 5/x (negative exponent)
• 2√x (fractional exponent)
• 3x⁻² (negative exponent)
• x + y (sum of terms)

Understanding the Degree of a Monomial

The degree of a monomial is the sum of the exponents of its variables.

Examples:

• 5x² has a degree of 2.
• -3y has a degree of 1 (remember, y¹ = y).
• 7 has a degree of 0 (it's a constant).
• x³y²z has a degree of 3 + 2 + 1 = 6.

What are Polynomials?

A polynomial is a sum or difference of monomials. Each monomial within a polynomial is called a term.

Examples of Polynomials:

• 3x² + 5x - 2 (this is a trinomial – a polynomial with three terms)
• 7y⁴ - 2y² + y + 9 (this is also a polynomial with four terms)
• -x⁵ + 4x³ - 6x (this polynomial has three terms)
• 8 (this is a constant polynomial, a monomial considered a polynomial with one term)

Examples that are NOT Polynomials:

• 1/x + 2 (contains a negative exponent)
• √x - 5 (contains a fractional exponent)
• 2ˣ (variable in the exponent)

The Degree of a Polynomial

The degree of a polynomial is the highest degree among its terms.

Examples:

• 3x² + 5x - 2 has a degree of 2.
• 7y⁴ - 2y² + y + 9 has a degree of 4.
• -x⁵ + 4x³ - 6x has a degree of 5.

Types of Polynomials Based on the Number of Terms

• Monomial: One term (e.g., 5x²)
• Binomial: Two terms (e.g., 2x + 3)
• Trinomial: Three terms (e.g., x² + 2x - 1)

Adding and Subtracting Polynomials

Adding and subtracting polynomials involves combining like terms. Like terms are terms with the same variables raised to the same powers.

Example: Adding Polynomials

Add (3x² + 2x - 5) + (x² - 4x + 7)

1. Group like terms: (3x² + x²) + (2x - 4x) + (-5 + 7)
2. Combine like terms: 4x² - 2x + 2

Example: Subtracting Polynomials

Subtract (2x³ - x + 3) - (x³ + 4x - 1)

1. Distribute the negative sign: 2x³ - x + 3 - x³ - 4x + 1
2. Group like terms: (2x³ - x³) + (-x - 4x) + (3 + 1)
3. Combine like terms: x³ - 5x + 4

Multiplying Monomials

Multiplying monomials involves multiplying the coefficients and adding the exponents of like variables.

Example:

(3x²)(2x⁴) = (3 * 2)(x² * x⁴) = 6x⁶

Multiplying Polynomials

Multiplying polynomials requires using the distributive property (also known as the FOIL method for binomials).

Example: Multiplying a Monomial by a Polynomial

2x(x² + 3x - 4) = 2x(x²) + 2x(3x) + 2x(-4) = 2x³ + 6x² - 8x

Example: Multiplying Binomials (FOIL Method)

(x + 2)(x - 3)

• F (First): x * x = x²
• O (Outer): x * -3 = -3x
• I (Inner): 2 * x = 2x
• L (Last): 2 * -3 = -6

Combine like terms: x² - 3x + 2x - 6 = x² - x - 6

Example: Multiplying Polynomials of Higher Degree

For polynomials with more than two terms, you need to use the distributive property repeatedly, ensuring that each term in one polynomial multiplies each term in the other. This often involves a tabular method or organization to avoid missing terms. Consider this example:

(x² + 2x - 1)(x + 3)

You can use a table:

Adding the terms together: x³ + 5x² + 5x - 3

Working with Polynomial Expressions

Often, you'll encounter more complex polynomial expressions that require a combination of the techniques we've covered. For example:

Simplify: 2(x + 3)² - 3x(x - 1) + 5

1. Expand the squared binomial: 2(x² + 6x + 9) - 3x(x - 1) + 5
2. Distribute: 2x² + 12x + 18 - 3x² + 3x + 5
3. Combine like terms: -x² + 15x + 23

Practice Problems

To solidify your understanding, try these practice problems:

1. Identify which expressions are monomials and which are polynomials: a) 4x³y² b) 2x⁻¹ + 5 c) 7x² - 3x + 1 d) √x

2. Determine the degree of each polynomial: a) 5x⁴ - 2x² + x - 7 b) 3y³ + 8 c) 6

3. Add the polynomials: (2x² + 5x - 3) + (x² - 2x + 1)

4. Subtract the polynomials: (3x³ - x² + 2) - (x³ + 2x² - 1)

5. Multiply the monomials: (-4a³b)(2ab²)

6. Multiply the polynomials: (x + 4)(x - 2)

7. Multiply the polynomials: (2x² - x + 3)(x + 2)

8. Simplify the expression: 3(x - 2)(x + 1) - 2x(x - 5)

These exercises will help you reinforce your understanding of the concepts discussed. Remember to show your work step-by-step, and don't hesitate to review the examples provided if you get stuck. Mastering these fundamentals is crucial for success in the remaining parts of Unit 5. Focus on understanding the underlying principles rather than just memorizing procedures. By practicing consistently and seeking clarification when needed, you'll build a strong foundation in polynomial algebra.