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Dividing Polynomials
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Division by a Monomial = 5xy x ≠ 0 y ≠ 0 = - 9a2 b a ≠ 0 b ≠ 0 = x2 - 2x + 3 x ≠ 0
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Basic Long Division - A Review
Quotient 5 7 8 8 4 0 6 3 Dividend Divisor 5 6 7 1 6 4 7 Remainder 4631 = 8 x Dividend = Divisor • Quotient + Remainder
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Division of a Polynomial by a Binomial
Dividend = Divisor x Quotient + Remainder P(x) = D(x) Q(x) + R(x) This is called the Division Statement or Division Algorithm. Divide: x2 + 7x + 2 ÷ (x + 2) 1. The polynomial must be in descending order of powers. Any missing terms are to be filled with a zero placeholder. Multiply x + 5 x + 2 x2 + 7x + 2 x2 + 2x x(x + 2) = x2 + 2x 5x + 2 2. Only the first term is used when doing the division. 5x + 10 Divide x2 x -8 = x 3. Multiply your answer with the entire divisor. 4. Subtract, bring down the next term and repeat the process. P(x) = (x + 2) (x + 5) - 8
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Division by a Binomial 2m2 + m - 4 2m + 1 4m3 + 4m2 - 7m - 3 4m3 + 2m2
Multiply Multiply Multiply 2m2 + m - 4 2m + 1 4m3 + 4m2 - 7m - 3 4m3 + 2m2 2m2 - 7m 2m2 + m - 8m - 3 - 8m - 4 1 P(x) = D(x) Q(x) + R(x) P(m) = (2m + 1)(2m2 + m - 4) + 1
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Division by a Binomial 4x2 - 3x + 2 x - 2 4x3 - 11x2 + 8x + 10 4x x2 - 3x2 + 8x - 3x2 + 6x 2x + 10 2x 14 P(x) = D(x) Q(x) + R(x) P(x) = (x - 2)(4x2 - 3x + 2) + 14
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Division by a Binomial Divide: (8m3 - 1) ÷ (2m - 1) 4m2 + 2m + 1 2m - 1 8m3 + 0m2 + 0m - 1 8m3 - 4m2 4m2 + 0m 4m2 - 2m 2m - 1 2m - 1 P(m) = (2m - 1)(4m2 + 2m + 1)
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Using Synthetic Division
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5 1 -2 -33 90 5 15 -90 3 -18 Quotient Rem Synthetic Division 1
Divide x3 - 2x x by (x - 5) using synthetic division. 5 1 -2 -33 90 1. Write only the constant term of the divisor, and the coefficients of the dividend. Add Add Add 5 15 Multiply Multiply -90 Multiply 1 3 -18 2. Bring down the first term of the dividend. Quotient Rem 3. Multiply 1 by 5, record the product and add. Written as x2 + 3x - 18 4. Multiply 3 by 5, record the product and add. Using the division statement: P(x) = (x - 5)(x2 + 3x - 18) 5. Multiply -18 by 5, record the product and add.
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Using Synthetic Division
Divide: (x4 - 2x3 + x2 + 12x - 6) ÷ (x - 2) 2 2 2 28 1 1 14 22 P(x) = D(x) Q(x) + R(x) What does this mean ???? P(x) = (x - 2) (x3 + x + 14) + 22 Remainder Divisor Quotient
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Using Synthetic Division
1. (4x3 - 11x2 + 8x + 6) ÷ (x - 2) P(x) = (x - 2)(4x2 - 3x + 2) + 10 8 -6 4 4 -3 2 10 2. (2x3 - 2x2 + 3x + 3) ÷ (x - 1) P(x) = (x - 1)(2x2 + 3) + 6
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The Remainder Theorem Given f(x) = x3 - 4x2 + 5x + 1, determine
the remainder when f(x) is divided by x - 1. 1 The remainder is 3. 1 -3 2 1 -3 2 3 NOTE: f(1) gives the same answer as the remainder using synthetic division. Using f(x) = x3 - 4x2 + 5x + 1, find f(1): f(1) = (1)3 - 4(1)2 + 5(1) + 1 = = 3 Therefore f(1) is equal to the remainder. In other words, when the polynomial x3 - 4x2 + 5x + 1 is divided by x - 1, the remainder is f(1).
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The Remainder Theorem:
When a polynomial f(x) is divided by x - b, the remainder is f(b). [Think x - b, then x = b.] Find the remainder when x3 - 4x2 + 5x - 1 is divided by: a) x - 2 b) x + 1 Find f(2): f(2) = (2)3 - 4(2)2 + 5(2) - 1 = = 1 Find f(-1): f(-1) = (-1)3 - 4(-1)2 + 5(-1) - 1 = = -11 The remainder is 1. The remainder is -11.
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Divide: (6x3 - 7x2 + 10x + 6) ÷ (2x - 1) 1 2 6 -7 10 6
Dividing by ax - b Divide: (6x3 - 7x2 + 10x + 6) ÷ (2x - 1) 1 2 To use synthetic division, the coefficient of x in the divisor must be 1. Therefore, you factor the divisor: 3 -2 4 6 -4 8 10 P(x) = (x - 1/2)(6x2 - 4x + 8) + 10 Note that the divisor is P(x) = (x - 1/2)(2)(3x2 - 2x + 4) + 10 Note that the quotient has a factor of 2 in it. Multiply the factor with the divisor. P(x) = (2x - 1)(3x2 - 2x + 4) + 10
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In general I find that it is easier to do
Dividing by ax - b Divide: (6a3 + 4a2 + 9a + 6) ÷ (3a + 2) The divisor is -2 3 -4 -6 6 9 P(x) = (a + 2/3)(6a2 + 9) P(x) = (a + 2/3)(3)(2a2 + 3) Factor out the 3 from the quotient. P(x) = 3(a + 2/3)(2a2 + 3) Multiply the 3 through the divisor. P(x) = (3a + 2)(2a2 + 3) In general I find that it is easier to do problems like the last two using long division!
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Finding the Remainder When Dividing by ax - b
Find the remainder when P(x) = 2x3 + x2 + 5x - 1 is divided by 2x - 1. For the divisor 2x - 1, factor out the a to determine the value of b. P(1/2) = 2(1/2)3 + (1/2)2 + 5(1/2) - 1 = 2(1/8) +(1/4) + 5/2 - 1 = 1/4 + 1/4 + 5/2 - 1 = 4 1 + 1 + 10 - 4 = 2 The remainder is 2.
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Factor Theorem If f(x) is a polynomial function, then
(x – c) is a factor of f(x) if and only if f(c) = 0. Ex: Given f(x) = x2 + 3x – 18, (x + 6) is a factor of f(x) if and only if f(-6) = 0. (-6)2 + 3(-6) – 18 = 0 Ex: Given f(x) = x2 + 3x – 18, (x - 3) is a factor of f(x) if and only if f(3) = 0. (3)2 + 3(3) – 18 = 0 We can check these results by using synthetic division.
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You can also use synthetic division to find factors of a polynomial...
Example: Given that (x + 2) is a factor of f(x), factor the polynomial f(x) = x3 – 13x2 + 24x + 108 Since (x + 2) is a factor, –2 is a zero of the function… We can use synthetic division to find the other factors... – – –2 30 –108 1 –15 54 This means that you can write x3 – 13x2 + 24x = (x + 2)(x2 – 15x + 54) This is called the depressed polynomial Factor this = (x + 2)(x – 9)(x – 6) The complete factorization is (x + 2)(x – 9)(x – 6)
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