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Polynomials appear throughout mathematics and the sciences. Once you understand the fundamentals, the operations—adding, subtracting, multiplying and dividing—become routine. While division can be a bit more involved, the core techniques are straightforward and reliable.

Polynomials: Definition and Examples

A polynomial is an algebraic expression that contains one or more terms with variables, integer exponents, and constants. Key constraints:

• No division by a variable.
• No negative or fractional exponents.
• Only a finite number of terms.

Examples:

\(x^3 + 2x^2 – 9x – 4\)

\(xy^2 – 3x + y\)

Polynomials can be categorized by degree (the highest total exponent) or by the number of terms: monomials (1 term), binomials (2 terms), trinomials (3 terms), etc.

Adding and Subtracting Polynomials

To combine polynomials, group like terms—terms that share the same variables and exponents. Coefficients can differ.

Example: Combine (x^3 + 3x) + (9x^3 + 2x + y)

Step 1 – group like terms:

\((x^3 + 9x^3) + (3x + 2x) + y\)

Step 2 – add coefficients:

\(10x^3 + 5x + y\)

For subtraction, distribute the minus sign and then combine like terms.

Example: (4x^4 + 3y^2 + 6y) – (2x^4 + 2y^2 + y)

Rewrite:

\(4x^4 + 3y^2 + 6y – 2x^4 – 2y^2 – y\)

Combine:

\((4x^4 – 2x^4) + (3y^2 – 2y^2) + (6y – y) = 2x^4 + y^2 + 5y\)

When a minus sign precedes a bracket, remember to flip the sign of each term inside.

Example: (4xy + x^2) – (6xy – 3x^2)

Expands to:

\(4xy + x^2 – 6xy + 3x^2\)

Multiplying Polynomial Expressions

Use the distributive property: multiply every term of the first polynomial by every term of the second, then combine like terms.

Example: 4x × (2x^2 + y)

\(4x × 2x^2 + 4x × y = 8x^3 + 4xy\)

More complex:

\((2y^3 + 3x) \times (5x^2 + 2x)\)

\(= (2y^3 \times 5x^2) + (2y^3 \times 2x) + (3x \times 5x^2) + (3x \times 2x)\)

\(= 10y^3x^2 + 4y^3x + 15x^3 + 6x^2\)

Dividing Polynomial Expressions

Long division follows the same pattern as numerical long division. Write the divisor on the left and the dividend on the right.

Example: \frac{x^2 – 3x – 10}{x + 2}

Step 1 – divide the leading terms: x^2 ÷ x = x. Write x above the line.

Step 2 – multiply: x(x + 2) = x^2 + 2x. Subtract from the dividend:

x^2 – 3x – 10 minus x^2 + 2x = –5x – 10.

Step 3 – bring down the next term (here, –10). Repeat:

Divide leading terms: (–5x) ÷ x = –5. Multiply: –5(x + 2) = –5x – 10.

Subtract: (–5x – 10) – (–5x – 10) = 0. No remainder.

Result: x – 5.

Whenever possible, factoring the dividend before division can simplify the process.

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