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Polynomial Addition: Like Terms

CombineLike Terms To add polynomials, we must Combine Like Terms Say we want to add these two polynomials: x 2 - 3x + 4 and 5x 2 - 2x - 2 x 2 + 5x 2 6x 2 Like Terms have exactly the same variables with exactly the same powers. Combine Like Terms by adding or subtracting the coefficients, but keep the variables (and powers) the same. (Use the sign rules of integers to determine whether to add or subtract) These terms both have x 2, so they are like terms

CombineLike Terms To add polynomials, we must Combine Like Terms Now add the next set of like terms: x 2 - 3x + 4 and 5x 2 - 2x x - 2x 6x 2 Like Terms have exactly the same variables with exactly the same powers. Combine Like Terms by adding or subtracting the coefficients, but keep the variables (and powers) the same. (Use the sign rules of integers to determine whether to add or subtract) - 5x - 5x These terms each have an x, so they are like terms

CombineLike Terms To add polynomials, we must Combine Like Terms Now add the last set of like terms: x 2 - 3x + 4 and 5x 2 - 2x x 2 Like Terms have exactly the same variables with exactly the same powers. Answer: 6x 2 - 5x x - 5x These terms are constants (numbers with no variables), so they are like terms

Horizontal Method

One method of adding polynomials is called the Horizontal Method Add 2x 2 - x - 7 and -x 2 + 3x - 4, use the Horizontal Method: Write the second polynomial in parentheses with a plus sign between them 2x 2 - x ( -x 2 + 3x - 4) + 1 Distribute the +1 2x 2 - x x 2 +3x- 4

2x 2 - x 2 x2x2x2x2 Combine Like Terms by adding or subtracting the coefficients, but keep the variables (and powers) the same. (Use the sign rules of integers to determine whether to add or subtract) These terms both have x 2, so they are like terms 2x 2 - x x 2 + 3x - 4 Horizontal Method: Horizontal Method: Polynomial Addition Now add the first set of like terms:

Horizontal Method: Horizontal Method: Polynomial Addition Now add the next set of like terms: 2x 2 - x x 2 + 3x - 4 x2x2x2x2 -x + 3x These terms each have an x, so they are like terms + 2x + 2x

x2x2x2x2 Answer: x 2 + 2x x + 2x These terms are constants (numbers with no variables), so they are like terms Horizontal Method: Horizontal Method: Polynomial Addition Now add the next set of like terms: 2x 2 - x x 2 + 3x - 4

Practice Problems: (Hit enter to see the answers) Add using the Horizontal Method 1) -6x 2 + 2x + 1 and 3x 2 - x + 2 5) 5x + 2x - 3 and 4x + 2 2) 5xy + 4x and -3xy - 12x 6) -3y 2 + 2y and y 2 + y - 1 3) 4ab + 2a 2 b and 3ab 7) 2xy - 5x and - 3xy + 6x - 7 4) 3x 2 y +4x 3 y and - x 3 y + 2x 2 y 8) -17x + 6 and 3x - 6 Answers: 1) -3x 2 + x + 3 2) 2xy - 8x 3) 2a 2 b + 7ab 4) 3x 3 y + 5x 2 y 5) 11x - 1 6) -2y 2 + 3y - 1 7) -xy + x - 7 8) -14x

Vertical Method

Add 4x 2 + 3x - 6 and 2x 2 - 5x + 4, use the Vertical Method : 4x 2 + 2x 2 Write the two polynomials so that the like terms are stacked on top of each other These terms both have x 2, so they are like terms Vertical Method: Vertical Method: Polynomial Addition

4x 2 + 2x 2 Write the two polynomials so that the like terms are stacked on top of each other These terms both have an x, so they are like terms + 3x - 5x Add 4x 2 + 3x - 6 and 2x 2 - 5x + 4, use the Vertical Method : Vertical Method: Vertical Method: Polynomial Addition

4x 2 + 2x 2 Write the two polynomials so that the like terms are stacked on top of each other These terms are constants, so they are like terms + 3x - 5x Add 4x 2 + 3x - 6 and 2x 2 - 5x + 4, use the Vertical Method : Vertical Method: Vertical Method: Polynomial Addition

Now draw a line under the whole thing and add the coefficients. 4x 2 + 2x 2 + 3x - 5x x 2 - 2x - 2 ANSWER = Add 4x 2 + 3x - 6 and 2x 2 - 5x + 4, use the Vertical Method : Vertical Method: Vertical Method: Polynomial Addition

Add x and 6x 2 - 5x - 3, use the Vertical Method : Write the two polynomials so that the like terms are stacked on top of each other These terms both have x 2, so they are like terms x 2 + 6x 2 Vertical Method: Vertical Method: Polynomial Addition

Add x and 6x 2 - 5x - 3, use the Vertical Method : x 2 + 6x 2 + 0x - 5x (no x term?) Solution: Write in a zero where there are missing terms. (Or you can leave a blank spot) A problem that comes up when using the Vertical Method is that sometimes there are terms missing. Vertical Method: Vertical Method: Polynomial Addition

Add x and 6x 2 - 5x - 3, use the Vertical Method : Write the two polynomials so that the like terms are stacked on top of each other These terms are constants, so they are like terms x 2 + 6x x - 5x Vertical Method: Vertical Method: Polynomial Addition

Now draw a line under the whole thing and add the coefficients. 7x 2 - 5x - 1 ANSWER = - 5x x 2 + 6x 2 + 0x Add x and 6x 2 - 5x - 3, use the Vertical Method : Vertical Method: Vertical Method: Polynomial Addition

Suggestions for other situations: SituationSolution 1. A term has no coefficient showingWrite a “1” in front of it Example: x 2 + 3x + 1 1x 2 + 3x There are more than two like terms Stack (or group) all like terms together Ex: 2x + 6x - 3 and 4x + 5 (2x + 6x + 4x) + (-3 + 5) 3. There are many missing terms Write in zeros for each of them Ex: 5x 3 - 2x and 4x 4 + 3x 2 + x - 60x 4 + 5x 3 + 0x 2 - 2x + 0 4x 4 + 0x 3 + 3x 2 + 1x - 6 4x 4 + 5x 3 + 3x 2 + 1x Subtraction problemDistribute the (-1) before working the problem. x 2 + 3x (2x 2 + 6x - 2) x 2 + 3x x 2 - 6x + 2

Practice Problems: (Hit enter to see the answers) Add using the Vertical Method 1)-6x 2 + 2x + 1 and 3x 2 - x + 2 5) 5x + 2x - 3 and 4x + 2 2) 5y + 4x and -3y - 12x 6) -3y 2 + 2y and y 2 + y - 1 3) 4ab + 2a 2 and 3ab 7) 2xy - 5x and - 3xy + 6x - 7 4) 3x 2 y +4xy and - xy + 2x 2 y 8) -17x and 3x - 6 Answers: 1) -3x 2 + x + 3 2) 2y - 8x 3) 7ab + 2a 2 4) 5x 2 y+ 3xy 5) 11x - 1 6) -2y 2 + 3y - 1 7) -xy + x - 7 8) -17x 2 + 3x

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