MODELING ADDITION OF POLYNOMIALS Algebra tiles can be used to model polynomials. + – + – + – 1 –1 x –x x 2 –x 2 These 1-by-1 square tiles have an area of 1 square unit. These 1-by-x rectangular tiles have an area of x square units. These x-by-x rectangular tiles have an area of x 2 square units.

MODELING ADDITION OF POLYNOMIALS You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1. 1 Form the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1 with algebra tiles. x 2 + 4x + 2 + + + + + + + 2 x 2 – 3x – 1 + + – – – –

+ = MODELING ADDITION OF POLYNOMIALS You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1. 2 To add the polynomials, combine like terms. Group the x 2-tiles, the x-tiles, and the 1-tiles. x 2 + 4x + 2 + 2x 2 – 3x – 1 + – + + + + + + + + + – = + – – –

+ = MODELING ADDITION OF POLYNOMIALS You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1. 2 To add the polynomials, combine like terms. Group the x 2-tiles, the x-tiles, and the 1-tiles. x 2 + 4x + 2 + 2x 2 – 3x – 1 3 Find and remove the zero pairs. + – + The sum is 3x 2 + x + 1. + + + + + + + + – = + – – –

Adding and Subtracting Polynomials An expression which is the sum of terms of the form a x k where k is a nonnegative integer is a polynomial. Polynomials are usually written in standard form. Standard form means that the terms of the polynomial are placed in descending order, from largest degree to smallest degree. Polynomial in standard form: 2 x 3 + 5x 2 – 4 x + 7 Leading coefficient Degree Constant term The degree of each term of a polynomial is the exponent of the variable. The degree of a polynomial is the largest degree of its terms. When a polynomial is written in standard form, the coefficient of the first term is the leading coefficient.

Classified by number of terms Classifying Polynomials A polynomial with only one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Identify the following polynomials: Classified by degree Classified by number of terms Polynomial Degree 6 constant monomial –2 x 1 linear monomial 3x + 1 1 linear binomial –x 2 + 2 x – 5 2 quadratic trinomial 4x 3 – 8x 3 cubic binomial 2 x 4 – 7x 3 – 5x + 1 4 quartic polynomial

(5x 3 – x + 2 x 2 + 7) + (3x 2 + 7 – 4 x) + (4x 2 – 8 – x 3) Adding Polynomials Find the sum. Write the answer in standard format. (5x 3 – x + 2 x 2 + 7) + (3x 2 + 7 – 4 x) + (4x 2 – 8 – x 3) SOLUTION Vertical format: Write each expression in standard form. Align like terms. 5x 3 + 2 x 2 – x + 7 3x 2 – 4 x + 7 – x 3 + 4x 2 – 8 + 4x 3 + 9x 2 – 5x + 6

Horizontal format: Add like terms. Adding Polynomials Find the sum. Write the answer in standard format. (2 x 2 + x – 5) + (x + x 2 + 6) SOLUTION Horizontal format: Add like terms. (2 x 2 + x – 5) + (x + x 2 + 6) = (2 x 2 + x 2) + (x + x) + (–5 + 6) = 3x 2 + 2 x + 1

Subtracting Polynomials Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 No change –2 x 3 + 5x 2 – x + 8 – –2 x 3 + 3x – 4 + Add the opposite 2 x 3 – 3x + 4

Subtracting Polynomials Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 –2 x 3 + 5x 2 – x + 8 – –2 x 3 + 3x – 4 + 2 x 3 – 3x + 4 5x 2 – 4x + 12

Subtracting Polynomials Find the difference. (3x 2 – 5x + 3) – (2 x 2 – x – 4) SOLUTION Use a horizontal format. (3x 2 – 5x + 3) – (2 x 2 – x – 4) = (3x 2 – 5x + 3) + (–1)(2 x 2 – x – 4) = (3x 2 – 5x + 3) – 2 x 2 + x + 4 = (3x 2 – 2 x 2) + (– 5x + x) + (3 + 4) = x 2 – 4x + 7

Using Polynomials in Real Life You are enlarging a 5-inch by 7-inch photo by a scale factor of x and mounting it on a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches less than twice as high as the enlarged photo. Write a model for the area of the mat around the photograph as a function of the scale factor. SOLUTION Use a verbal model. Area of photo Verbal Model Area of mat = Total Area – 7x 14x – 2 Area of mat = A (square inches) Labels 5x Total Area = (10x)(14x – 2) (square inches) 10x Area of photo = (5x)(7x) (square inches) …

Using Polynomials in Real Life You are enlarging a 5-inch by 7-inch photo by a scale factor of x and mounting it on a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches less than twice as high as the enlarged photo. Write a model for the area of the mat around the photograph as a function of the scale factor. SOLUTION … 7x A = (10x)(14x – 2) – (5x)(7x) 14x – 2 Algebraic Model = 140x 2 – 20x – 35x 2 5x = 105x 2 – 20x 10x A model for the area of the mat around the photograph as a function of the scale factor x is A = 105x 2 – 20x.