Polynomials—expressions with multiple terms, constants, variables, and exponents—are foundational in algebra. Understanding their structure lets you locate graph intercepts, solve equations, and analyze functions.

Finding the Degree of a Polynomial

Step 1: Identify the Highest Exponent

For -9x⁶ – 3, the variable is x and the highest power is 6, so the degree is 6.

Step 2: Choose the Largest Exponent When Multiple Terms Exist

In 8x⁹ – 7x³ + 2x² – 9, the largest exponent of x is 9, making the degree 9.

Step 3: Add Exponents in Multivariable Polynomials

For 4x³y² – 3x²y⁴, add the exponents of each variable: x (3 + 2 = 5) and y (2 + 4 = 6). The overall degree is 6.

Simplifying Polynomials

Step 1: Combine Like Terms (Addition)

Combine (4x² – 3x + 2) + (6x² + 7x – 5) to get 10x² + 4x – 3.

Step 2: Distribute a Negative Sign (Subtraction)

Subtract (2x² – 7x – 3) from (5x² – 3x + 2) by distributing the negative, then combine like terms to obtain 3x² + 4x + 5.

Step 3: Apply the Distributive Property (Multiplication)

Multiply 4x(3x² + 2) to get 12x³ + 8x.

Factoring Polynomials

Step 1: Extract the Greatest Common Factor (GCF)

From 15x² – 10x, factor out 5x to obtain 5x(3x – 2).

Step 2: Use Grouping for Higher‑Degree Polynomials

Rewrite 18x³ – 27x² + 8x – 12 as two groups: (18x³ – 27x²) + (8x – 12). Factor each group, then factor out the common binomial (2x – 3) to arrive at (2x – 3)(9x² + 4).

Step 3: Factor a Perfect‑Square Trinomial

Identify x² – 22x + 121 as a square of (x – 11) because 11² = 121. Verify by expanding: (x – 11)(x – 11) = x² – 22x + 121.

Solving Equations by Factoring

Step 1: Apply the Zero Product Property

Set 4x³ + 6x² – 40x = 0 equal to zero.

Step 2: Factor Stepwise

Factor out 2x: 2x(2x² + 3x – 20) = 0, then factor the trinomial: 2x(2x – 5)(x + 4) = 0.

Step 3: Solve Each Factor

• 2x = 0 → x = 0
• 2x – 5 = 0 → x = 5/2
• x + 4 = 0 → x = –4

These are the three solutions to the cubic equation.