# Adding, Subtracting, and Multiplying Polynomials

π§Zayn Alazawi πJuly 5th, 2020 π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: $$(3x^2β2x+5)+(2x^2+4xβ1)$$

Solution: Combine like terms: $$(3x^2+2x^2)+(β2x+4x)+(5β1)$$$$5x^2+2x+4$$

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: $$(3x^2β2x+5)β(2x^2+4xβ1)$$

Solution: Distribute the negative sign: $$3x ^2 β2x+5β2x^2 β4x+1$$

Combine like terms:

$$(3x ^2 β2x ^2 )+(β2xβ4x)+(5+1)$$

$$x ^2 β6x+6$$

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: $$(3xβ2)(2x+1)$$

Solution: Use the distributive property: $$3x(2x)+3x(1)β2(2x)β2(1)$$

Combine like terms:

$$6x ^2 +3xβ4xβ2$$

$$6x ^2 βxβ2$$

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

## βοΈPractice

1. Evaluate:

$$(2xβ3)(3x+1)$$

2. Apply the distributive property:

$$2x(3x)+2x(1)β3(3x)β3(1)$$

Combine like terms:

$$6x ^2 +2xβ9xβ3$$

$$6x^2β7xβ3$$

3. Evaluate:

$$(4x ^2 β7x+3)β(2x ^2 β4x+2)$$

4. Distribute the negative sign:

$$4x ^2 β7x+3β2x ^2 +4xβ2$$

Combine like terms:

$$(4x ^2 β2x ^2 )+(β7x+4x)+(3β2)$$

$$2x ^2 β3x+1$$