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**Addition, Subtraction, and Multiplication of Polynomials**

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

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Polynomials A polynomial in one variable x is an expression which is the sum of only constant terms, x-terms, x2-terms, x3-terms, etc. For example, 7x4 – 2x2 + 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 y3-terms and x2y4-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 3x2 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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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. (3x2 – 2x + 1) + (7x3 – 2x2 + 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 = 3x2 – 2x x3 – 2x2 + x + 1 Combining like terms we get: = 7x3 + x2 – x + 2 Next Slide

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Practice Problems Add the polynomials 4x4 – 2x2 + x + 7 and 3x3 – 2x + 8. Click for answers: 4x4 + 3x3 2x2 – x + 15 Next Slide

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**Subtraction of Polynomials**

Example. (7x2 + 2x – 1) – (3x2 – 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 3x2, subtract 2x, and subtract 8. = 7x2 + 2x – 1 – 3x2 + 2x – 8 Note that the signs changed on each term in the second polynomial. = 4x2 + 4x – 9. Example. (5x2 – 2x + 7) – (10x3 – 7x2 + 8x – 13) = 5x2 – 2x + 7 – 10x3 + 7x2 – 8x + 13 = 10x3 + 12x2 – 10x + 20. Next Slide

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**Practice Problems Subtract: (x2 + 2x – 3) – (7x – 4).**

Subtract: (x2 – 4) – (3x3 – 2x2 + x – 7). Answers: 1) x2 – 5x + 1 2) 3x3 + 3x2 – x + 3 Next Slide

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**Multiplying Polynomials**

We’ve done a little of this, but not too much. Consider the multiplication (x2 – 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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**Example In (x2 – 2x + 2)(x + 1), each of the terms x2, 2x, and 2**

must be multiplied times each of the terms x and 1. Let’s be systematic: x2 times x and x2 times 1 2x times x and 2x times 1 2 times x and 2 times 1. This gives us the new terms: x3 and x2 2x2 and 2x 2x and 2 These get added together: x3 + x2 – 2x2 – 2x + 2x + 2 = x3 – x = x3 – x2 + 2 Next Slide

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Example (3x2 – x)(x2 – 2x + 3) 3x2 times everything = (3x2)(x2) + (3x2)(-2x) + (3x2)(3) x times everything + (-x)(x2) + (-x)(-2x) + (-x)(3) = 3x4 – 6x3 + 9x2 – x3 + 2x2 – 3x = 3x4 – 7x3 + 11x2 – 3x. Next Slide

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**Example (x2 + 2x – 1)(2x2 – 3x + 2) = (x2)(2x2) + (x2)(-3x) + (x2)(2)**

Next Slide

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**Practice Problems (x2 + 3)(x – 2) (x2 + 2x – 1)(x + 7)**

Answers: 1) x3 – 2x2 + 3x – 6 2) x3 + 9x2 + 13x – 7 3) x4 + x3 – 3x2 + 7x – 6 End

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