Dividing Polynomials

Rational Expressions A rational expression is a mathematical expression containing polynomials in the numerator and denominator. In all rational expressions, the denominator can not be equal to zero, since division by zero is undefined. Therefore, values of the variable that would make the denominator equal to zero must be listed. These values are called Restricted Values.They should be listed for every rational expression.

Listing Restricted Values x ≠ 0 x ≠ -3 x ≠ 0, -3 x ≠ 0, ±3 x ≠ 4, - 5 x ≠ ±4 xy ≠ 0 (3a - b) ≠ 0 (3a + b) ≠ 0 x ≠ 0 (x - y) ≠ 0 (x + y) ≠ 0

Division by a Monomial = 5xy x ≠ 0 y ≠ 0 = - 9a2 b a ≠ 0 b ≠ 0 = x2 - 2x + 3 x ≠ 0

Basic Long Division - A Review Quotient 5 7 8 8 4 6 3 1 4 0 6 3 Dividend Divisor 5 6 7 1 6 4 7 Remainder 4631 = 8 x 578 + 7 Dividend = Divisor • Quotient + Remainder

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

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

Division by a Binomial 4x2 - 3x + 2 x - 2 4x3 - 11x2 + 8x + 10 4x3 - 8x2 - 3x2 + 8x - 3x2 + 6x 2x + 10 2x - 4 14 P(x) = D(x) Q(x) + R(x) P(x) = (x - 2)(4x2 - 3x + 2) + 14

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)

Using Synthetic Division

Quotient Divisor Dividend Remainder + = The Division Statement Quotient Divisor Dividend Remainder 66 = (7) X (9) + 3 + Dividend = Divisor X Quotient Remainder For any division problem: Dividend = (Divisor)(Quotient) + Remainder P(x) = D(x) • Q(x) + R(x)

5 1 -2 -33 90 5 15 -90 3 -18 Quotient Rem Synthetic Division 1 Divide x3 - 2x2 - 33x + 90 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.

Using Synthetic Division Divide: (x4 - 2x3 + x2 + 12x - 6) ÷ (x - 2) 2 1 -2 1 12 -6 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

Using Synthetic Division 1. (4x3 - 11x2 + 8x + 6) ÷ (x - 2) 2 4 -11 8 6 P(x) = (x - 2)(4x2 - 3x + 2) + 10 8 -6 4 4 3 2 10 2. (2x3 - 2x2 + 3x + 3) ÷ (x - 1) 1 2 -2 3 3 2 0 3 P(x) = (x - 1)(2x2 + 3) + 6 2 0 3 6

The Remainder Theorem Given f(x) = x3 - 4x2 + 5x + 1, determine the remainder when f(x) is divided by x - 1. 1 1 -4 5 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 = 1 - 4 + 5 + 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).

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 = 8 - 16 + 10 - 1 = 1 Find f(-1): f(-1) = (-1)3 - 4(-1)2 + 5(-1) - 1 = -1 - 4 - 5 - 1 = -11 The remainder is 1. The remainder is -11.

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: 6 -7 10 6 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

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 6 4 9 6 -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!

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.

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.

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 1 –13 24 108 –2 30 –108 1 –15 54 This means that you can write x3 – 13x2 + 24x + 108 = (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)