Adding, Subtracting, and Multiplying Polynomials

Let's explore the fundamental operations of adding, subtracting, and multiplying polynomials.

I. Adding Polynomials:

Definition: Adding polynomials involves combining like terms. Like terms are terms with the same variable(s) raised to the same exponent(s).

Procedure:

1. Identify like terms in each polynomial.
2. Combine the coefficients of like terms.

Example: \((3x^2−2x+5)+(2x^2+4x−1)\)

Solution: Combine like terms: \((3x^2+2x^2)+(−2x+4x)+(5−1)\); \(5x^2+2x+4\)

II. Subtracting Polynomials:

Definition: Subtracting polynomials is similar to addition. However, when subtracting, distribute the negative sign to all terms in the second polynomial.

Procedure:

1. Distribute the negative sign to all terms in the second polynomial.
2. Follow the steps for adding polynomials.

Example: \((3x^2−2x+5)−(2x^2+4x−1)\)

Solution: Distribute the negative sign: \(3x ^2 −2x+5−2x^2 −4x+1\)

Combine like terms:

\((3x ^2 −2x ^2 )+(−2x−4x)+(5+1)\)

\(x ^2 −6x+6\)

III. Multiplying Polynomials:

Definition: Multiplying polynomials involves applying the distributive property multiple times.

Procedure:

1. Multiply each term in the first polynomial by each term in the second polynomial.
2. Combine like terms.

Example: \((3x−2)(2x+1)\)

Solution: Use the distributive property: \(3x(2x)+3x(1)−2(2x)−2(1)\)

Combine like terms:

\(6x ^2 +3x−4x−2\)

\(6x ^2 −x−2\)

These operations are crucial in polynomial manipulation and finding solutions to various mathematical problems.