Warm-Up. Use long division to divide 5 into

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1 Warm-Up Use long division to divide 5 into

2 Warm-Up Use long division to divide 5 into Divisor Quotient Dividend Remainder

3 Warm-Up Use long division to divide 5 into Dividend Divisor Quotient Remainder Divisor

4 Remainders This means that the divisor is a factor of the dividend If you are lucky enough to get a remainder of zero when dividing, then the divisor divides evenly into the dividend For example, when dividing 3 into 192, the remainder is 0. Therefore, 3 is a factor of 192.

5 Objective 1a You will be able to divide polynomials using long division

6 Dividing Polynomials Dividing polynomials works just like long division. In fact, it is called long division! Before you start dividing: Make sure the divisor and dividend are in standard form If your polynomial is missing a term, add it in with a coefficient of 0 as a place holder

7 Dividing Polynomials Dividing polynomials works just like long division. In fact, it is called long division! Before you start dividing: 2x 3 + x + 3 2x 3 + 0x 2 + x + 3 If your polynomial is missing a term, add it in with a coefficient of 0 as a place holder

8 Exercise 1 Divide x + 1 into x 2 + 3x + 5 x 2 2 x 1 x 3x x - x 2x 5-2 x How many times does x go into x 2? Multiply x by x + 1 Multiply 2 by x + 1 Line up the first term of the quotient with the term of the dividend with the same degree.

9 Exercise 1 Divide x + 1 into x 2 + 3x + 5 x 2 2 x 1 x 3x x - x 2x 5-2 x - 2 Divisor 3 Quotient Dividend Remainder

10 Exercise 1 Divide x + 1 into x 2 + 3x + 5 Dividend x 2 3x 5 3 x 2 x 1 x 1 Remainder Divisor Divisor Quotient

11 Exercise 2 Divide 3x 4 5x 3 + 4x 6 by x 2 3x + 5

12 Exercise 3 In a polynomial division problem, if the degree of the dividend is m and the degree of the divisor is n, what is the degree of the quotient?

13 Exercise 4 Divide using long division. 1. x 3 x 2 +4x 10 x x 4 +x 3 +x 1 x 2 +2x 1

14 Exercise 4 Divide using long division. 1. x 3 x 2 +4x 10 x+2

15 Exercise 4 Divide using long division. 2. 2x 4 +x 3 +x 1 x 2 +2x 1

16 Objective 1b You will be able to divide polynomials using synthetic division

17 Exercise 5 Use long division to divide x 4 10x 2 + 2x + 3 by x 3

18 Synthetic Division When you divisor is of the form x k, where k is a constant, then you can perform the division quicker and easier using just the coefficients of the dividend. This is called fake division. I mean, synthetic division.

19 Synthetic Division Synthetic Division (of a Cubic Polynomial) To divide ax 3 + bx 2 + cx + d by x k, use the following pattern. k a b c d = Add terms a ka Coefficients of Quotient (in decreasing order) = Multiply by k Remainder

20 Synthetic Division Synthetic Division (of a Cubic Polynomial) To divide ax 3 + bx 2 + cx + d by x k, use the following pattern. k a b c d = Add terms a ka = Multiply by k You are always adding columns using synthetic division, whereas you subtracted columns in long division.

21 Synthetic Division Synthetic Division (of a Cubic Polynomial) To divide ax 3 + bx 2 + cx + d by x k, use the following pattern. Add a coefficient of zero for any missing terms! k can be positive or negative. If you divide by x + 2, then k = -2 because x + 2 = x (-2). You are always adding columns using synthetic division, whereas you subtracted columns in long division.

22 Exercise 6 Use synthetic division to divide x 4 10x 2 + 2x + 3 by x 3

23 Exercise 7 Divide 2x 3 + 9x 2 + 4x + 5 by x + 3 using synthetic division

24 Exercise 8 Divide using long division. 1. x 3 +4x 2 x 1 x x 3 +x 2 3x+7 x 1

25 Exercise 9 Given that x 4 is a factor of x 3 6x 2 + 5x + 12, rewrite x 3 6x 2 + 5x + 12 as a product of two polynomials.

26 Exercise 10 The volume of the solid is 3x 3 + 8x 2 45x 50. Find an expression for the missing dimension. x + 5?

27 Exercise 10 The volume of the solid is 3x 3 + 8x 2 45x 50. x + 5?

28 Exercise 11 Use long division to divide 6x 4 11x x 2 3x 1 by 2x 1. Then figure out a way to perform the division synthetically.

29 Exercise 11 Use long division to divide 6x 4 11x x 2 3x 1 by 2x 1. Then figure out a way to perform the division synthetically.

30 5.5ish: Divide Polynomials Objectives: 1. To divide polynomials using long division and synthetic division

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