Chapter 5-4: Dividing Polynomials

St. Augustine Preparatory SchoolOctober 5, 2015

Last Class (Ch. 5-1, pg 288)- Finding zeros of a polynomial function

y = (x+2)(x-1)(x-3) has zeros: -2, 1, 3- Writing a polynomial from its zeros

if a polynomial has zeros: -2, 2, 3, then the polynomial is y = (x+2)(x-2)(x-3), which then multiplies out to be y = x3 – 3x2 – 4x +12- Relative maximum or relative minimum

- R. Max: value of the function at an up-to-down turning point

- R. Min: Value of the function at a down-to-up turning point

Relative maximum/minimum

Factoring When the variable with the highest exponent down not have a constant of 1 (ie. 9x3 + 6x2 - 3x)

Step 1: Write down the functionStep 2: Factor out 3x from each termStep 3: Find two numbers that multiply to be -3x2 and add together to be 2x. Insert them into the equation. Step 4: Factor again if necessaryStep 5: Use the term before each of the identical binomials to make anew binomial and multiply it by the binomial that both terms had in common.

Chapter 5-4: Dividing Polynomials

• Similar to long division

Polynomial Long DivisionExample: 4x2 + 23x – 16 divided by x+5

Checking factors using long division

• You can use long division to figure out if one function is a factor of another. If the two functions are divided and the remainder = 0, then the divisor is a factor of the function. If it does not = 0, it is not a factor.

• Continue using the long division process until the degree of the remainder is less than the degree of the divisor.

Checking factors using long division

Checking factors using long division

• Example: Is x2 + 1 a factor of 3x4 – 4x3 + 12x2 + 5?***Do not forget to add in the terms that are

missing***This means: x2 + 1 should be wrote as x2 + 0x + 1 and 3x4 – 4x3 + 12x2 + 5 must be rewrote to include a 0x term as well.

Answer: no it is not a factor

Assignment Questions

• Page 308: 9, 13, 14, 15, 18, 20

If finished, read through problem 3 on how to do synthetic division

Synthetic Division

Step 1: Reverse the sign of the constant in the expression that is being divided by. Write the coefficients of the polynomial to the right of it.Step 2: Bring down the first coefficientStep 3: Multiply the coefficient by the the divisor. Add it to the next coefficient in lineStep 4: Continue until all terms have been added by. The last term is the remainder.

Example

Divide x3 – 14x2 + 51x -54 by x+2

Another example

Divide x3 – 57x + 56 by x- 7. What is the quotient and the remainder?

Quotient: x2 + 7x – 8 Remainder: 0

Answer: x2 + 7x – 8, R 0

Remainder Theorem

• If you divide a polynomial P(x) of degree n ≥ 1 by x – a, then the remainder is P(a)

• Example: If you divide P(x) = x5 – 2x3 – x2 + 2, by x – 3, then the remainder is equal to P(3)

P(3) = 35 – 2(3)3 – (3)2 + 2P(3) = 182

The remainder is 182

Remainder Theorem

• Note: this only works when dividing by x – a. If you have a polynomial that is x + a, remember that it could be wrote as x – (-a).

• Ex. If you were to divide P(x) by:1. x – 4, you would find P(4)2. x + 3, you would find P(-3)3. x + 9, you would find P(-9)4. x – 1, you would find P(1)

Assignment Questions

• Page 308, 21, 24, 25, 28, 34, 37, 38