Adding and Subtracting Polynomials Notes

Learning Objectives for Adding and Subtracting Polynomials

• Add and subtract polynomials using algebra tiles.

• Add and subtract polynomials by combining like terms.

• Solve problems involving the addition and subtraction of polynomials.

The Distributive Property

The distributive property is an algebraic property used to multiply a single value by two or more values contained within a set of parentheses. The formulas for this property are as follows:

• $A(B + C) = AB + AC$

• $A(B - C) = AB - AC$

Steps for Adding and Subtracting Monomials

To perform operations with monomials, follow these specific steps:

1. Check if the monomials are like terms: To be considered like terms, the monomials must share the same variable and the same exponent.

2. Add or subtract the numerical coefficients: Perform the operation on the numbers in front of the variables.

3. Maintain the variable and exponent: Keep the variable and the exponent the same as they were in the original terms; do not change them during addition or subtraction.

Examples of Monomial Addition

• $12x + 6x = 18x$

• $-8m^2 + 3m^2 = -5m^2$

• $9d^2 + 4d = 9d^2 + 4d$ (Note: These are not like terms because they have different exponents.)

• $7b^3 + 2b^3 - 6b^2 = 9b^3 - 6b^2$

• $-10v + 4v + 18v = 12v$

Adding Polynomials by Combining Like Terms

Polynomials can be added using a vertical method or a horizontal method. The primary goal is to remove grouping symbols and group similar terms together.

Example: $3x + 2 + (4x + 6)$

Vertical Solution:

$3x + 2$ $+ (4x + 6)$ $\text{----------}$ $7x + 8$

Horizontal Solution:

1. Remove the parentheses: $3x + 2 + 4x + 6$

2. Combine like terms (group numbers with numbers and variables with variables): $3x + 4x + 2 + 6$

3. Simplify: $7x + 8$

Example: $x^2 + 2 + (2x^2 - 5)$

Vertical Solution:

$x^2 + 2$ $+ (2x^2 - 5)$ $\text{----------}$ $3x^2 - 3$

Horizontal Solution:

1. Remove the parentheses and apply the distributive property: $x^2 + 2 + 2x^2 - 5$

2. Combine like terms: $x^2 + 2x^2 - 5 + 2$

3. Simplify: $3x^2 - 3$

Example: $3x^2 - 8x + 7 + (2x^2 - 6x + 2)$

Vertical Solution:

$3x^2 - 8x + 7$ $+ (2x^2 - 6x + 2)$ $\text{----------}$ $5x^2 - 14x + 9$

Horizontal Solution:

1. Remove the parentheses from the first polynomial and apply the distributive property to the second polynomial: $3x^2 - 8x + 7 + 2x^2 - 6x + 2$

2. Group similar terms and simplify: $3x^2 + 2x^2 - 8x - 6x + 2 + 7$

3. Final Result: $5x^2 - 14x + 9$

Examples of Monomial Subtraction

• $12x - 6x = 6x$

• $-8m^2 - 3m^2 = -11m^2$

• $9d^2 - 4d = 9d^2 - 4d$

• $7b^3 - 2b^3 - 6b^2 = 5b^3 - 6b^2$

• $-10v - 4v - 18v = -32v$

Subtracting Polynomials

When subtracting polynomials, the distributive property is applied to the second polynomial, which reverses the sign of every term inside the parentheses.

Example: $3x^2 + 5 - (x^2 - 1)$

Vertical Solution:

$3x^2 + 5$ $-(x^2 - 1)$ $\text{----------}$ $2x^2 + 6$

Horizontal Solution:

1. Remove parentheses from the first polynomial: $3x^2 + 5$

2. Apply the distributive property to the second polynomial (change signs): $-x^2 + 1$

3. Combine terms: $3x^2 - x^2 + 5 + 1$

4. Simplify: $2x^2 + 6$

Example: $7x^2 + 36 - (5x^2 + x + 9)$

Vertical Solution:

Note: A placeholder of $0x$ is used for the missing term in the first polynomial.

$7x^2 + 0x + 36$ $-(5x^2 + x + 9)$ $\text{----------}$ $2x^2 - x + 27$

Horizontal Solution:

1. Expand the expression by distributing the negative sign: $7x^2 + 36 - 5x^2 - x - 9$

2. Group similar terms: $7x^2 - 5x^2 - x + 36 - 9$

3. Simplify: $2x^2 - x + 27$

Example: $9a^3 + 5a^2 + 11a - (-2a^2 + 8a^3 - 9a)$

Vertical Solution:

$9a^3 + 5a^2 + 11a$ $-(8a^3 - 2a^2 - 9a)$ $\text{----------}$ $a^3 + 7a^2 + 20a$

Horizontal Solution:

1. Remove parentheses and apply distributive property to the second polynomial: $9a^3 + 5a^2 + 11a + 2a^2 - 8a^3 + 9a$

2. Group similar terms: $9a^3 - 8a^3 + 5a^2 + 2a^2 + 9a + 11a$

3. Simplify: $a^3 + 7a^2 + 20a$

Adding and Subtracting Polynomials using Algebra Tiles

Algebra tiles provide a visual representation of polynomial operations.

Representation Legend

• $x^2$ Tile: Represents the squared variable term.

• $x$ Tile: Represents the linear variable term.

• Unit Tile: Represents constant values.

• Positive Tile: Typically represented by a specific color or sign to indicate positive values.

• Negative Tile: Typically represented by a specific color or sign to indicate negative values.

Zero Pair

A zero pair is a pair of tiles that add up to zero. Every zero pair consists of one positive tile and one negative tile of the same type.

Examples using Algebra Tiles

Example 1: $2x + 3x$

• Represent $2x$ with two positive $x$ tiles.

• Represent $3x$ with three positive $x$ tiles.

• Combining them results in five positive $x$ tiles.

• Result: $5x$

Example 2: $10n^2 - 4n^2$

• This problem can be rewritten as $10n^2 + (-4n^2)$.

• Represent $10n^2$ with ten positive $n^2$ tiles.

• Represent $-4n^2$ with four negative $n^2$ tiles.

• Match the four negative tiles with four of the positive tiles to create zero pairs.

• Remove the zero pairs.

• The remaining six positive $n^2$ tiles represent the solution.

• Result: $6n^2$

Example 3: $(2x^2 + x + 1) + (3x^2 - x + 5)$

• Step 1: Represent both polynomials with tiles.

  • Polynomial 1: Two positive $x^2$ tiles, one positive $x$ tile, and one positive unit tile.

  • Polynomial 2: Three positive $x^2$ tiles, one negative $x$ tile, and five positive unit tiles.

• Step 2: Group similar tiles together.

  • Combine the $x^2$ tiles: $2x^2 + 3x^2$

  • Combine the $x$ tiles: $x - x$

  • Combine the unit tiles: $5 + 1$

• Step 3: Identify and remove zero pairs.

  • The single positive $x$ tile and the single negative $x$ tile form a zero pair and are removed.

• Step 4: Count the remaining tiles.

  • Five positive $x^2$ tiles remain.

  • Six positive unit tiles remain.

• Result: $5x^2 + 6$