Assignment, red pen, pencil, highlighter, GP notebook M3D6 Have out: Bellwork: State the degree of each function, identify the leading coefficient, and find the zeros: 1) y = x3 – 4x2 2) y = 2x2 – 10x + 12 x3 – 4x2 = 0 2x2 – 10x + 12 = 0 2(x2 – 5x + 6) = 0 x2(x – 4) = 0 2(x – 2)(x – 3) = 0 x2 = 0 x – 4 = 0 x – 2 = 0 x – 3 = 0 x = 0 x = 4 x = 2 x = 3 Degree:________ Leading coefficient:________ Zeros:________ 3 Degree:________ Leading coefficient:________ Zeros:________ 2 +1 +2 x = 0, 0, 4 x = 2, 3

Factoring Higher Degree Trinomials Factoring higher degree trinomials is very similar to the method we use when factoring quadratics Let’s review what we have done since Algebra 1: Example: Factor the quadratic 8 2 4 6

Example: Factor the trinomial Trinomials such as can also be written in quadratic form as Ex: These trinomials can be factored as the product of two binomials. Split x4 into x2 and x2. These will be the first terms in the binomials. 8 6 Ex: 2 4 If you are unsure of your answer, use F.O.I.L. or a generic rectangle to check your answers.

Practice: a) 5 6 b) -18 -3 1 5 -6 3 c) 12 -8 -6 -2

Recall from yesterday: Even degree + leading coefficient Even degree – leading coefficient Odd degree + leading coefficient Odd degree – leading coefficient

Or remember it this way: Even Degree, raise the roof!

Odd Degree, Saturday Night Fever!

Graphing Polynomial Functions, Part 2 Determine the following information for each polynomial function. Graph the function. 1. y = (x – 4)(x + 3) 0 = (x – 4)(x + 3) x – 4 = 0 x + 3 = 0 x = 4 x = –3 Be sure to plot the y–intercept, too! x = -3, 4 zeros: _____________ (0, –12) 2 +1 degree: ____ leading coefficient: an = ________ Endpoint behavior: (choose one)

2. y = –x(x – 3)(x + 3) 0 = –x(x – 3)(x + 3) –x = 0 x – 3 = 0 x + 3 = 0 x = 0 x = 3 x = –3 Odd degree, so down then up. Wait, the L.C. is negative! x = –3, 0, 3 zeros: _____________ end behavior: ________ 3 –1 degree: ____ leading coefficient: an = ________

zeros: _____________ end behavior: ________ 3. y = (x + 3)2(x + 1) (0, 9) 0 = (x + 3)2(x + 1) (x + 3)2 = 0 x + 1 = 0 x + 3 = 0 x = –1 x = –3 x = –3, –3, –1 zeros: _____________ end behavior: ________ 3 +1 degree: ____ leading coefficient: an = ________ double tangent With a _______ factor, the polynomial is ________ to the x–axis at the _____________. x–intercepts

Try exercises #4 on your own.

4. y = x(x + 1)(x – 2) 0 = x(x + 1)(x – 2) x = 0 x + 1 = 0 x – 2 = 0 x = –1 x = 2 zeros: _____________ end behavior: ________ x = –1, 0, 2 3 1 degree: ____ leading coefficient: an = ________

5. y = –2x2(x – 1) (x + 1) –2x2 = 0 x – 1 = 0 x + 1 = 0 x = 0 x = 1 x = –1 Bounce! x = –1, 0, 0, 1 zeros: _____________ end behavior: ________ 4 –2 degree: ____ leading coefficient: an = ________ Skip #6 and #7 for right now. You need to finish these tonight. Let’s go to #8.

8. y = x(x – 3)3 x = 0 (x – 3)3 = 0 x – 3 = 0 x = 3 triple zero! At this zero, the graph will curve like a cubic. 0, 3, 3, 3 zeros: _____________ end behavior: ________ 4 +1 degree: ____ leading coefficient: an = ________

zeros: _____________ end behavior: ________ 6. y = (x – 3)2(x + 1)2 (x – 3)2 = 0 (x + 1)2 = 0 x – 3 = 0 x + 1= 0 (0, 9) x = 3 x = –1 Bounce! x = –1, –1, 3, 3 zeros: _____________ end behavior: ________ 4 1 degree: ____ leading coefficient: an = ________

zeros: _____________ end behavior: ________ 7. y = –2(x – 2)(x + 3)2(x – 1) x –2 = 0 (x + 3)2 = 0 x – 1 = 0 x = 2 x + 3 = 0 x = 1 x = –3 Bounce! (0, –36) x = –3, –3, 1, 2 zeros: _____________ end behavior: ________ 4 –2 degree: ____ leading coefficient: an = ________

Summary: When we graph a polynomial function, 1. Determine and graph the _________. (y = __ ) ( _________ if necessary.) 2. Graph the _________. (x = __ ) 3. Find and graph the ________ behavior 4. Remember, the polynomial function is __________ to the x–axis for any ________ zeros. 5. Graph the rest of the polynomial. zeros Factor y–intercept endpoint tangent double

Old slides

6. y = (x – 3)2 + 2 locator point: 0 = (x – 3)2 + 2 (3, 2) –2 = (x – 3)2 No solution?! We’ll have to use our “imagination” to address this later. None zeros: _____________ end behavior: ________ 2 +1 degree: ____ leading coefficient: an = ________

Fractions Add to your notes: Baby Math Review How to multiply fractions: Multiply straight across. 2 6 How to divide fractions: Flip the second fraction. 5 1 Change the sign to multiplication. Multiply across.

Rational Expressions Add to your notes: Take out the rational expressions worksheet. Let’s do problems #2 and #8. 3 3 We can simplify since everything is multiplied. 2) 2 4 Multiply across and simplify.

Rational Expressions Add to your notes: 8) Flip the second fraction. Change the sign to multiplication. (m + 1)(m + 1) 10(m – 1) 2 Simplify and multiply across.