Dividing Polynomials

Simple Division - dividing a polynomial by a monomial

Simplify

Simplify

Long Division - divide a polynomial by a polynomial Think back to long division from 3rd grade. How many times does the divisor go into the dividend? Put that number on top. Multiply that number by the divisor and put the result under the dividend. Subtract and bring down the next number in the dividend. Repeat until you have used all the numbers in the dividend.

x - 8 x2/x = x -8x/x = -8 -( ) + 3x x2 - 8x - 24 -( ) - 8x - 24

h2 + 4h + 5 4h2/h = 4h 5h/h = 5 h3/h = h2 -( ) - 4h2 h3 - 11h 4h2 -( ) 4h2 - 16h 5h + 28 -( ) 5h - 20 48

Synthetic Division - To use synthetic division: divide a polynomial by a polynomial To use synthetic division: There must be a coefficient for every possible power of the variable. The divisor must have a leading coefficient of 1.

Step #1: Write the terms of the. polynomial so the degrees are in Step #1: Write the terms of the polynomial so the degrees are in descending order. Since the numerator does not contain all the powers of x, you must include a 0 for the

5 -4 1 6 Since the divisor is x-3, r=3 Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients. 5 -4 1 6 Since the divisor is x-3, r=3

Step #3: Bring down the first coefficient, 5.

15 15 5 Step #4: Multiply the first coefficient by r, so and place under the second coefficient then add. 5 15 15

Step #5: Repeat process multiplying. the sum, 15, by r; Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add. 5 15 45 41

Step #5 cont.: Repeat the same procedure. Where did 123 and 372 come from? 5 15 45 41 123 372 124 378

Step #6: Write the quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend. 5 15 45 41 123 124 372 378

The quotient is: Remember to place the remainder over the divisor.

Ex 7: Step#1: Powers are all accounted for and in descending order. Step#2: Identify r in the divisor. Since the divisor is x+4, r=-4 .

4 -4 20 8 -5 -1 1 -2 10 Step#3: Bring down the 1st coefficient. Step#4: Multiply and add. Step#5: Repeat. 4 -4 20 8 -5 -1 1 -2 10

Ex 8: Notice the leading coefficient of the divisor is 2 not 1. We must divide everything by 2 to change the coefficient to a 1.

3

*Remember we cannot have complex fractions - we must simplify.

Ex 9: 1 Coefficients

Remainder and Factor Theorems

REMAINDER THEOREM -2 2 -3 2 -1 -4 14 -32 2 -7 16 -33 Let f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c). Let’s look at an example to see how this theorem is useful. So the remainder we get in synthetic division is the same as the answer we’d get if we put -2 in the function. The root of x + 2 = 0 is x = -2 using synthetic division let’s divide by x + 2 -2 2 -3 2 -1 -4 14 -32 2 -7 16 -33 the remainder Find f(-2)

Opposite sign goes here FACTOR THEOREM Let f be a polynomial function. Then x – c is a factor of f (x) if and only if f (c) = 0 If and only if means this will be true either way: 1. If f(c) = 0, then x - c is a factor of f(x) 2. If x - c is a factor of f(x) then f(c) = 0. Try synthetic division and see if the remainder is 0 12 -51 153 -3 -4 5 0 8 -4 17 -51 161 Opposite sign goes here NO it’s not a factor. In fact, f(-3) = 161 We could have computed f(-3) at first to determine this. Not = 0 so not a factor