Unit 4 Polynomial Review- Key Concepts and Practice Problems

Unit 4 Polynomial Review: Everything You Need to Know

Polynomials show up everywhere in algebra, and Unit 4 is where things get serious. If you're struggling with this unit, you're not alone—but reading this once won't fix it. You actually have to practice.

This guide cuts through the fluff and gives you the concepts, methods, and problems you need to actually understand polynomials. No motivational quotes. Just math.

What Is a Polynomial?

A polynomial is an expression with constants, variables, and exponents combined using addition, subtraction, and multiplication. That's it.

Examples:

• 3x² + 2x - 5
• 4x³ - x² + 7x + 1
• 2y - 9

Constants are just numbers. Variables are letters that represent unknown values. Exponents tell you how many times to multiply the variable by itself.

Key Vocabulary

• Term – a single piece separated by + or - signs
• Coefficient – the number in front of the variable
• Degree – the highest exponent in the polynomial
• Leading coefficient – the coefficient of the highest-degree term
• Constant term – the term with no variable

For 4x³ - x² + 7x + 1: the degree is 3, the leading coefficient is 4, and the constant term is 1.

Types of Polynomials

Polynomials are classified by two things: how many terms they have and their degree.

Classification by Number of Terms

• Monomial – one term (5x³)
• Binomial – two terms (x² + 4)
• Trinomial – three terms (2x² - 3x + 7)

Classification by Degree

Anything degree 5 or higher? Just call it a "degree 5 polynomial." Nobody uses special names past quartic.

Operations with Polynomials

Adding and Subtracting Polynomials

Combine like terms. Like terms have the same variable raised to the same power.

Example:

(3x² + 2x - 5) + (x² - 3x + 7)

= 3x² + x² + 2x - 3x - 5 + 7

= 4x² - x + 2

For subtraction, distribute the negative sign first. Then combine like terms.

Multiplying Polynomials

Multiplying a monomial by a polynomial: Use the distributive property.

2x(3x² - 4x + 5) = 6x³ - 8x² + 10x

Multiplying two binomials: Use FOIL (First, Outer, Inner, Last).

(x + 3)(x - 2)

= x² - 2x + 3x - 6

= x² + x - 6

Multiplying larger polynomials: Multiply every term in the first polynomial by every term in the second. Then combine like terms.

Dividing Polynomials

You have two options: long division and synthetic division.

Long division works for all polynomials. Synthetic division only works when dividing by a linear expression (x - c), but it's faster.

Factoring Polynomials

Factoring is breaking a polynomial into simpler pieces that multiply back together. This is essential for solving equations.

Greatest Common Factor (GCF)

Find what factor every term shares.

6x³ + 9x² - 3x

The GCF is 3x.

= 3x(2x² + 3x - 1)

Factoring Trinomials (x² + bx + c)

Find two numbers that multiply to c and add to b.

x² + 5x + 6

Two numbers that multiply to 6 and add to 5: 2 and 3

= (x + 2)(x + 3)

Factoring a² - b² (Difference of Squares)

a² - b² = (a + b)(a - b)

x² - 16 = (x + 4)(x - 4)

Factoring a³ - b³ (Difference of Cubes)

a³ - b³ = (a - b)(a² + ab + b²)

x³ - 8 = (x - 2)(x² + 2x + 4)

Factoring a³ + b³ (Sum of Cubes)

a³ + b³ = (a + b)(a² - ab + b²)

x³ + 27 = (x + 3)(x² - 3x + 9)

Polynomial Theorems

Remainder Theorem

When you divide f(x) by (x - c), the remainder equals f(c). That's it.

Quick way to check if (x - c) is a factor: plug c into the polynomial. If you get 0, (x - c) is a factor.

Factor Theorem

A polynomial f(x) has a factor (x - c) if and only if f(c) = 0.

This follows directly from the Remainder Theorem. When the remainder is 0, you have a factor.

Rational Root Theorem

If a polynomial has rational roots, they follow this pattern:

Possible rational roots = ±(factors of constant term) ÷ (factors of leading coefficient)

Example: f(x) = 2x² + 5x - 3

Constant term: 3 → factors: ±1, ±3

Leading coefficient: 2 → factors: ±1, ±2

Possible rational roots: ±1, ±3, ±1/2, ±3/2

Test these in the polynomial to find actual roots. Not all will work—that's normal.

Fundamental Theorem of Algebra

Every polynomial of degree n has exactly n complex roots (counting multiplicity).

A cubic always has 3 roots. A quartic always has 4 roots. Some might be repeated; some might be complex numbers.

Graphing Polynomials

Key things to identify on a polynomial graph:

• x-intercepts – where the graph crosses the x-axis (roots/zeros)
• y-intercept – where the graph crosses the y-axis (plug in x = 0)
• End behavior – what happens as x approaches ±∞

End Behavior Rules

Look at the leading term and whether the degree is even or odd:

Multiplicity and the Graph

Multiplicity is how many times a root appears.

• Even multiplicity (2, 4, 6...): graph touches the x-axis and bounces back
• Odd multiplicity (1, 3, 5...): graph crosses through the x-axis

How to Solve Polynomial Equations

Step 1: Set the polynomial equal to zero.

Step 2: Factor completely.

Step 3: Use the Zero Product Property—if ab = 0, then a = 0 or b = 0.

Step 4: Solve each factor for the variable.

Example: x² - 5x + 6 = 0

Factor: (x - 2)(x - 3) = 0

Set each equal to zero: x - 2 = 0 → x = 2

x - 3 = 0 → x = 3

Practice Problems

Try these. No peeking at the answers until you've attempted them.

Problem 1

Factor completely: x² - 9

Answer: (x + 3)(x - 3)

Problem 2

Factor: 2x² + 7x + 3

Answer: (2x + 1)(x + 3)

Problem 3

Divide using synthetic division: (x³ - 4x² + x + 6) ÷ (x - 2)

Answer: x² - 2x - 3, remainder 0

Problem 4

Solve: x³ - 4x² - 7x + 10 = 0

Answer: x = -2, x = 1, x = 5

Problem 5

Find all possible rational roots of: 3x³ - 2x² + 4x - 8

Answer: ±1, ±2, ±4, ±1/3, ±2/3, ±4/3

Problem 6

State the end behavior of: -2x⁴ + 3x² - 1

Answer: Falls on both ends (degree even, leading coefficient negative)

Quick Reference: Factoring Methods

Common Mistakes to Avoid

• Forgetting to check for a GCF first – always look for this before trying other methods
• Sign errors when distributing negatives – write it out step by step
• Not testing all possible rational roots – the first one you try usually isn't right
• Confusing end behavior rules – draw a quick sketch if you're unsure
• Forgetting multiplicity – a double root touches the axis; a single root crosses through

Final Thoughts

Polynomials aren't hard—they're just procedural. Every problem has a method. Learn the methods. Practice until you can do them without thinking.

Start with factoring. If you can't factor reliably, you'll struggle with everything else. Once factoring clicks, solving equations and graphing become much easier.

Do the practice problems. Then do more. There's no way around it.