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Polynomials

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A polynomial is a monomial or a sum of monomials. Binomial: sum of two monomials Trinomial: sum of three monomials

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Polynomials MonomialsBinomialsTrinomials 3x 8 5y-2X+2y-z -5xyz 2 4b 2 +6bX 2 +5x-7 5h5-6y4a 2 -2b+6

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Polynomials Degree of a monomial is the sum of the exponents of the variables. Degree of a polynomial is the greatest degree of any of its’ monomials. MonomialDegree 8y 3 3 6y 3 b 8 k12 180 Polynomial Degree of terms Degree of the polynomials 3x 2 +4xy 3 +2 2, 4, 0 4 7y 4 +3xy-8x 3 y 2 4, 2, 5 5

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Polynomials Ascending order goes up. Descending order goes down. Look at the exponent of the variable you are arranging. All the other variables exponents don’t matter. 5x 4 y 2 +8x 9 y 7 -19xy 3 +x 5 y

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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. Adding and Subtracting Polynomials Standard form means that the terms of the polynomial are placed in descending order, from largest degree to smallest degree. The degree of each term of a polynomial is the exponent of the variable. Polynomial in standard form: 2 x 3 + 5x 2 – 4 x + 7 DegreeConstant termLeading coefficient 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.

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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: Classifying Polynomials PolynomialDegree Classified by degree Classified by number of terms 6 –2 x 3x + 1 –x 2 + 2 x – 5 4x 3 – 8x 2 x 4 – 7x 3 – 5x + 1 0 1 1 4 2 3 constant linear quartic quadratic cubic monomial binomial polynomial trinomial binomial

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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 ) Adding Polynomials 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

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Find the sum. Write the answer in standard format. (2 x 2 + x – 5) + (x + x 2 + 6) Adding Polynomials 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

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Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) Subtracting Polynomials 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 + 3x – 4– Add the opposite No change –2 x 3 + 5x 2 – x + 8 2 x 3 – 3x + 4 +

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Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) Subtracting Polynomials 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 + 3x – 4– 5x 2 – 4x + 12 –2 x 3 + 5x 2 – x + 8 2 x 3 – 3x + 4 +

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

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Total Area = (10x)(14x – 2) (square inches) Area of photo = 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. Using Polynomials in Real Life Write a model for the area of the mat around the photograph as a function of the scale factor. Verbal Model Labels Area of mat = Area of photo Area of mat = A (5x)(7x) (square inches) Total Area – Use a verbal model. 5x5x 7x7x 14x – 2 10x SOLUTION …

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(10x)(14x – 2) – (5x)(7x) 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. Using Polynomials in Real Life Write a model for the area of the mat around the photograph as a function of the scale factor. A = = 140x 2 – 20x – 35x 2 SOLUTION = 105x 2 – 20x A model for the area of the mat around the photograph as a function of the scale factor x is A = 105x 2 – 20x. Algebraic Model … 5x5x 7x7x 14x – 2 10x

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