# Adding, Subtracting, and Multiplying Polynomials

🧑Zayn Alazawi 📅July 5th, 2020 📊Grade 11 Math

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

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.

## ✍️Practice

1. Evaluate:

$$(2x−3)(3x+1)$$

2. Apply the distributive property:

$$2x(3x)+2x(1)−3(3x)−3(1)$$

Combine like terms:

$$6x ^2 +2x−9x−3$$

$$6x^2−7x−3$$

3. Evaluate:

$$(4x ^2 −7x+3)−(2x ^2 −4x+2)$$

4. Distribute the negative sign:

$$4x ^2 −7x+3−2x ^2 +4x−2$$

Combine like terms:

$$(4x ^2 −2x ^2 )+(−7x+4x)+(3−2)$$

$$2x ^2 −3x+1$$