Opening Quiz: Sketch a graph of the following polynomial function by hand using the methods discussed on Friday (make sure to label and be specific as possible) : 𝑓 𝑥 = 𝑥 3 −9𝑥

Long Division of Polynomials If you're dividing a polynomial by something more complicated than just a simple monomial, then you'll need to use a different method for the simplification. That method is called "long (polynomial) division", and it works just like the long (numerical) division you did back in elementary school, except that now you're dividing with variables.

Long Division of Polynomials Divide 𝒙 𝟐 +𝟒𝒙−𝟖 by 𝒙−𝟐

Student Check: Answer the following: 𝒙 𝟑 +𝟒𝒙+𝟔 𝒙 𝟐 +𝟏

Try Again: Answer: −𝟐+𝟑𝒙−𝟓 𝒙 𝟐 +𝟒 𝒙 𝟑 +𝟐 𝒙 𝟒 𝒙 𝟐 +𝟐𝒙−𝟑

Class Opener: Solve the following problems using long division: (𝑥 3 +5 𝑥 2 −32𝑥−7) 𝑏𝑦 (𝑥−4) 𝑘 3 −30𝑘−18−4 𝑘 2 𝑏𝑦 (3+𝑘)

Synthetic Division There is a nice shortcut for long division of polynomials when dividing by divisors of the for (x – k). This short cut is known as SYNTHETIC DIVISION

Let’s look at how to do this using the example: In order to use synthetic division these two things must happen: #1 There must be a coefficient for every possible power of the variable. #2 The divisor must have a leading coefficient of 1.

Step #1: Write the terms of the polynomial so Step #1: Write the terms of the polynomial so the degrees are in descending order.

Step #2: Write the constant a of the divisor Step #2: Write the constant a of the divisor x- a to the left and write down the coefficients.

Step #3: Bring down the first coefficient, 5. Step #4: Multiply the first coefficient by r (3*5).

Step #5: After multiplying in the diagonals, add the column.

Add Multiply the diagonals, add the columns.

Step #7: Repeat the same procedure as step #6. Add Columns Add Columns Add Columns Add Columns

Step #8: 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.

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

Try this one:

Student Check Use synthetic division to divide 𝑥 4 −10 𝑥 2 −2𝑥+4 ÷𝑥+3

Exit Slip: 5 𝑥 3 +8 𝑥 2 −𝑥+6 𝑏𝑦 𝑥+2 5 𝑥 3 +18 𝑥 2 +7𝑥−6 𝑏𝑦 (𝑥+2) Use synthetic division to divide: 5 𝑥 3 +8 𝑥 2 −𝑥+6 𝑏𝑦 𝑥+2 5 𝑥 3 +18 𝑥 2 +7𝑥−6 𝑏𝑦 (𝑥+2)

Remainder Theorem If a polynomial f(x) is divided by x – k, the remainder is r = f(k)

Using Remainder Theorem: Use the Remainder Theorem to evaluate the following function at x = -2 𝑓 𝑥 =3 𝑥 3 +8 𝑥 2 +5𝑥−7

Factor Theorem: A polynomial f(x) has a factor (x-k) if and only if f(k) = 0

Factoring a Polynomial: Repeat Division Show that (x – 2) and (x+3) are factors of 𝑓 𝑥 =2 𝑥 4 +7 𝑥 3 −4 𝑥 2 −27𝑥−18 Then find the remaining factors of f(x)

Student Check: Show that (x -5) and (x+4) are factors of 𝑥 4 −4 𝑥 3 −15 𝑥 2 +58𝑥−40 Then find the remaining factors of f(x)

Using the Remainder in Synthetic Division: In summary, the remainder r, obtained in the synthetic division of f(x) by (x – k) provides the following information: If r = 0, (x – k) is a factor of f(x) The remainder, r, gives the value of f at x=k that is, r = f(k). If r = 0, (k,0) is an x-intercept of the graph of f

Rational Zero Test: The rational zero test relates the possible rational zeros of a polynomial (having integer coefficients) to the leading coefficient and to the constant term of the polynomial.

Using the Rational Zero Test: To us the rational zero test, first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. 𝑃𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑅𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑍𝑒𝑟𝑜𝑠= 𝐹𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑇𝑒𝑟𝑚 𝐹𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

Tips: If the leading coefficient is not 1, the list of possible zeros can increase dramatically. Use these tips to shrink your search: Use a graphing calculator to graph the polynomial Use IVT, and table function of calculator Use The Factor Theorem and synthetic division.

Using the Rational Zeros Test: Find the rational zeros of 𝑓 𝑥 =2 𝑥 3 +3 𝑥 2 −8𝑥+3

Student Check: Find the rational zeros of the following: 𝑥 3 +3 𝑥 2 −𝑥−3

Descartes’ Rule of Signs method of determining the maximum number of zeros in a polynomial. Look on page 124 in text.