Check: Worksheet 5.4 B (odds)

Test 5.1-5.5 Friday Algebra 2 Lesson 5.5 Quiz 5.4 tomorrow Check: Worksheet 5.4 B (odds)

Warm-up • Factor using the sum and difference of cubes formulas. • x3+8 2. 125x3-1 3. 54x3-128

Objective 1— Use long division and synthetic division. 2—Use the remainder theorem and the factor theorem.

Example 1: a) Use synthetic substitution to find f(4) if f(x) = x4 – 6x3 + 8x2 + 5x + 13 4 –8 0 20 33 1 –2 0 5 f(4) = 33 4 1 –6 8 5 13

b) f(x) = 2x4 + x3 – 3x2 – 5 Find f(2) AKA The Remainder Theorem 28 2 2 1 –3 0 –5 4 10 14 2 5 7 14 23 f(2) = 23

Synthetic division can be used when the divisor is in the form (x – k). k –16 8 4 Example 2: Use synthetic division for the followinga) (2x3– 7x2– 8x + 16) ÷ (x – 4) First, write down the coefficients in descending order, and k of the divisor in the form x – k : 4 2 –7 –8 16 These are the coefficients of the quotient (and the remainder) 2 1 –4 0 2x2 + x – 4

place-holder Notice that k is –1 since synthetic division works for divisors in the form (x – k). EX 2 (b) Divide (5x3 + x2 – 7) ÷ (x + 1) –5 4 –4 5 –4 4 –11 5x2 – 4x + 4 – 11 x + 1 –1 5 1 0 –7 Worksheet 5.5B Try #8, 10 Then we’ll continue with examples

You can also use synthetic division to find factors of a polynomial... Example 4: 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 30 –108 1 –15 54 0 –2 1 –13 24 108 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) Try #13, 14 Then we’ll continue with examples

Example 5: Find the other zeros of f given f(-2)=0 –2 1 8 5 –14 -2 -12 –14 0 1 6 -7 x2+6x-7=0 (x+7)(x-1)=0 Worksheet 5.5B Try #2, 24 Then we’ll continue with examples Zeros: {-7, 1, -2}

Rewrite in long division form... 2x + 7 divisor Think, how many times does x go into 2x2 ? dividend 2x2 – 4x Example 6:Divide using long division a)(2x2 + 3x – 4) ÷ (x – 2) Multiply by the divisor. 7x – 4 Subtract. 7x – 14 Think, how many times does x go into 7x ? 10 remainder (x – 2) 2x2 + 3x – 4 Write the result like this... 2x + 7 + 10 x – 2 divisor

p2 + p + 1 p3 – p2 Ex 1 (b): Divide (p3 – 6) ÷ (p – 1) Be sure to add “place-holders” for missing terms... p2 + 0p p2 – p p – 6 (p – 1) p3 + 0p2 + 0p – 6 p – 1 –5 p2 + p + 1 – 5 p – 1 Let’s look at an abbreviated form of long division, called synthetic division...

PRACTICEDivide using long division • Worksheet 5.5B • Do #4, 6 • together

Quiz 5.4 tomorrow Test 5.1-5.5 Friday Assignments • Classwork: Finish Worksheet 5.5B • Homework: p. 366 # 3-30(x3), 37,38 (12 pts)

Closure • The volume of the box is given by V = 2x3 - 11x2 + 10x + 8. Find an expression for the Missing dimension. 2x+1 x - 4

Check: Worksheet 5.4 B (odds)