Polynomial Problems- Practice and Solutions

What Are Polynomials?

A polynomial is a mathematical expression with variables and coefficients. You combine terms using addition, subtraction, and multiplication. No division by variables. No negative or fractional exponents.

The degree of a polynomial tells you the highest power of the variable. A degree-2 polynomial is quadratic. Degree-3 is cubic. Degree-4 is quartic. Get the pattern?

Most polynomial problems fall into a few categories: operations, factoring, and finding roots. Master these three and you're set.

Types of Polynomial Problems

Here's what you'll encounter:

• Simplifying polynomial expressions
• Adding and subtracting polynomials
• Multiplying polynomials (FOIL method for binomials)
• Dividing polynomials (long division and synthetic division)
• Factoring polynomials
• Finding zeros or roots
• Solving polynomial equations

Basic Polynomial Operations

Adding and Subtracting

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

Example:

(3x² + 2x + 5) + (4x² - 3x + 1) = 7x² - x + 6

Subtraction is the same, but distribute the negative sign first.

(5x³ + 3x²) - (2x³ - x²) = 5x³ + 3x² - 2x³ + x² = 3x³ + 4x²

Multiplying Polynomials

Use the distributive property. Multiply every term in the first polynomial by every term in the second.

For two binomials, FOIL works: First, Outer, Inner, Last.

Example:

(x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6

For larger polynomials, just multiply systematically. Don't try to skip steps until you're comfortable.

Factoring Polynomials

Factoring is reverse multiplication. You break a polynomial into products of simpler polynomials.

Factoring Out the GCF

Find the greatest common factor of all terms. Pull it out front.

Example:

6x³ + 9x² = 3x²(2x + 3)

The GCF here is 3x². That's it.

Factoring Trinomials

For x² + bx + c, find two numbers that multiply to c and add to b.

Example: x² + 5x + 6

Find two numbers that multiply to 6 and add to 5. That's 2 and 3.

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

Difference of Squares

a² - b² = (a + b)(a - b)

Example: x² - 9 = (x + 3)(x - 3)

Only works when you have a perfect square minus a perfect square. Not for sums.

Sum/Difference of Cubes

a³ + b³ = (a + b)(a² - ab + b²)

a³ - b³ = (a - b)(a² + ab + b²)

Example: x³ - 8 = x³ - 2³ = (x - 2)(x² + 2x + 4)

Finding Roots (Zeros)

The roots of a polynomial are the x-values that make the polynomial equal zero. Factor the polynomial, set each factor to zero, solve for x.

Example: x² - 5x + 6 = 0

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

Set each factor to zero: x - 2 = 0 → x = 2. x - 3 = 0 → x = 3.

Roots are x = 2 and x = 3.

The Rational Root Theorem

When you can't factor easily, use this. For a polynomial with integer coefficients, any rational root p/q must have p dividing the constant term and q dividing the leading coefficient.

Example: 2x² - 5x + 3 = 0

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

Test these. You'll find x = 1 and x = 3/2 work.

Dividing Polynomials

Polynomial Long Division

Works like regular long division. Divide the leading term of the dividend by the leading term of the divisor. Multiply, subtract, repeat.

Example: (x² + 5x + 6) ÷ (x + 2)

x² ÷ x = x. Multiply: x(x + 2) = x² + 2x. Subtract: (x² + 5x + 6) - (x² + 2x) = 3x + 6.

3x ÷ x = 3. Multiply: 3(x + 2) = 3x + 6. Subtract: 0.

Answer: x + 3

Synthetic Division

Use this when dividing by a linear binomial (x - c). It's faster.

Write the coefficients. Bring down the first coefficient. Multiply by c, add to next coefficient. Repeat.

Example: Divide x² + 5x + 6 by (x - 1)

c = 1. Coefficients: 1, 5, 6

Bring down 1. Multiply 1 × 1 = 1. Add: 5 + 1 = 6. Multiply 6 × 1 = 6. Add: 6 + 6 = 12.

Result: x + 6 with remainder 12. Or (x + 6) + 12/(x - 1)

Polynomial Division Methods Compared

Practice Problems with Solutions

Problem 1: Simplify (2x² + 3x - 1) + (x² - 2x + 4)

Solution: 3x² + x + 3

Problem 2: Multiply (x - 4)(x + 7)

Solution: x² + 3x - 28

Problem 3: Factor 2x² + 10x

Solution: 2x(x + 5)

Problem 4: Factor x² - 4x - 12

Solution: (x - 6)(x + 2)

Problem 5: Find the roots of x² - 9 = 0

Solution: (x + 3)(x - 3) = 0. Roots are x = 3 and x = -3.

Problem 6: Divide x³ + 2x² - 5x - 6 by (x - 1)

Solution using synthetic division: c = 1. Coefficients: 1, 2, -5, -6.

Bring down 1. Multiply 1 × 1 = 1. Add: 2 + 1 = 3. Multiply 3 × 1 = 3. Add: -5 + 3 = -2. Multiply -2 × 1 = -2. Add: -6 + (-2) = -8.

Quotient: x² + 3x - 2. Remainder: -8.

Common Mistakes to Avoid

• Forgetting to distribute the negative sign when subtracting polynomials
• Not combining like terms after expanding
• Factoring errors — always double-check by multiplying back
• Skipping the GCF when it's there
• Misapplying FOIL — it only works for binomial × binomial
• Forgetting that a² - b² factors but a² + b² does not (unless using complex numbers)

Getting Started with Polynomial Problems

Here's your approach:

1. Identify what the problem asks — simplify, factor, solve, or divide?
2. Look for common factors before doing anything else
3. Choose your method — factoring for quadratic equations, synthetic division for linear divisors
4. Check your work — multiply factors back out, plug answers into the original equation

Work through problems daily. Polynomials are mechanical — the more you practice, the faster you get.