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1 MA 1128: Lecture 10 – 10/13/14 Addition, Subtraction, and Multiplication of Polynomials

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2 Polynomials A polynomial in one variable x is an expression which is the sum of only constant terms, x-terms, x 2 -terms, x 3 -terms, etc. For example, 7x 4 – 2x 2 + x + 3 is a polynomial. The variable doesn’t have to be x, of course, and the book talks about polynomials in several variables. In several variables, we might have y 3 -terms and x 2 y 4 -terms. We’ll stick to one variable, for the most part. You could say that a “nomial” is the same as a term, so “polynomial” means many terms. We’ll actually take “poly” to mean any number of terms, so 3x 2 by itself would be a polynomial. Occasionally, the words monomial, binomial, and trinomial are used to explicitly describe a polynomial with one, two, or three terms. Next Slide

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3 Adding Polynomials We’ll need to be able to add and subtract polynomials. We’ve done this already, and called it simplifying. Pay special attention to combining like terms. Example. (3x 2 – 2x + 1) + (7x 3 – 2x 2 + x + 1). With addition, the parentheses don’t really do anything, since we can add the terms in any order. Dropping the parentheses, we get = 3x 2 – 2x x 3 – 2x 2 + x + 1 Combining like terms we get: = 7x 3 + x 2 – x + 2 Next Slide

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4 Practice Problems 1.Add the polynomials 4x 4 – 2x 2 + x + 7 and 3x 3 – 2x + 8. Click for answers: 4x 4 + 3x 3 2x 2 – x + 15 Next Slide

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5 Subtraction of Polynomials Example. (7x 2 + 2x – 1) – (3x 2 – 2x + 8). Here the second set of parentheses are doing something very important. They tell us that every term inside is being subtracted. We must subtract 3x 2, subtract 2x, and subtract 8. = 7x 2 + 2x – 1 – 3x 2 + 2x – 8 Note that the signs changed on each term in the second polynomial. = 4x 2 + 4x – 9. Example. (5x 2 – 2x + 7) – (10x 3 – 7x 2 + 8x – 13) = 5x 2 – 2x + 7 – 10x 3 + 7x 2 – 8x + 13 = 10x x 2 – 10x Next Slide

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6 Practice Problems 1.Subtract: (x 2 + 2x – 3) – (7x – 4). 2.Subtract: (x 2 – 4) – (3x 3 – 2x 2 + x – 7). Answers: 1) x 2 – 5x + 1 2) 3x 3 + 3x 2 – x + 3 Next Slide

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7 Multiplying Polynomials We’ve done a little of this, but not too much. Consider the multiplication (x 2 – 2x + 2)(x + 1). Recall that multiplication distributes over addition. A careful application of this concept leads to the fact that Every term in the first polynomial needs to be multiplied times every term in the second. Next Slide

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8 Example In (x 2 – 2x + 2)(x + 1), each of the terms x 2, 2x, and 2 must be multiplied times each of the terms x and 1. Let’s be systematic: x 2 times x and x 2 times 1 2x times x and 2x times 1 2 times x and 2 times 1. This gives us the new terms: x 3 and x 2 2x 2 and 2x 2x and 2 These get added together: x 3 + x 2 – 2x 2 – 2x + 2x + 2 = x 3 – x = x 3 – x Next Slide

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9 Example (3x 2 – x)(x 2 – 2x + 3) 3x 2 times everything= (3x 2 )(x 2 ) + (3x 2 )(-2x) + (3x 2 )(3) x times everything+ (-x)(x 2 ) + (-x)(-2x) + (-x)(3) = 3x 4 – 6x 3 + 9x 2 – x 3 + 2x 2 – 3x = 3x 4 – 7x x 2 – 3x. Next Slide

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10 Example (x 2 + 2x – 1)(2x 2 – 3x + 2) = (x 2 )(2x 2 ) + (x 2 )(-3x) + (x 2 )(2) + (2x)(2x 2 ) + (2x)(-3x) + (2x)(2) + (-1)(2x 2 ) + (-1)(-3x) + (-1)(2) = 2x 4 – 3x 3 + 2x 2 + 4x 3 – 6x 2 + 4x – 2x 2 + 3x – 2 = 2x 4 + x 3 – 6x 2 + 7x – 2 Next Slide

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11 Practice Problems 1.(x 2 + 3)(x – 2) 2.(x 2 + 2x – 1)(x + 7) 3.(x 2 – x + 2)(x 2 + 2x – 3) Answers: 1) x 3 – 2x 2 + 3x – 6 2) x 3 + 9x x – 7 3) x 4 + x 3 – 3x 2 + 7x – 6 End

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