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

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Presentation on theme: "1 MA 1128: Lecture 10 – 10/13/14 Addition, Subtraction, and Multiplication of Polynomials."— Presentation transcript:

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

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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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