1. 5.5, Day 2 More Synthetic Division

2. If 𝑓 𝑥 = 𝑥 3 + 𝑥 2 −16𝑥−16 has a zero of 4, what would the other zeros be?
If 4 is a zero, that means that x – 4 is a factor and I should be able to synthetically divide by 4 and find the other factors and from there find the other zeros.
Good, we expected a remainder of zero, and now we need to factor
𝑥 2 +5𝑥+4 . 4 20 16 1 5 4
𝑥 2 +5𝑥+4=(𝑥+1)(𝑥+4), so the other zeros are -1 and -4
What would f(-4) be for the above equation?
zero

3. If f(-1) = 0, completely factor 𝑓 𝑥 = 𝑥 3 +2 𝑥 2 −5𝑥−6.
If f(-1) = 0, then x + 1 is a factor, and I can synthetically divide by -1 and finish factoring. -1 -1 6 1 1 -6
𝑥 2 +𝑥−6=(𝑥+3)(𝑥−2)
Therefore the complete factorization of 𝑥 3 + 2𝑥 2 −5𝑥−6 is
𝑥+1 𝑥+3 𝑥−2 .

4. If f(-2) = 0, completely factor 𝑓 𝑥 = 7𝑥 3 −10 𝑥 2 −39𝑥+18.
-14 48
-18 7
-24 9
7 𝑥 2 −24𝑥+9=(7𝑥−3)(𝑥−3)
The complete factorization of 7𝑥 3 − 10𝑥 2 −39𝑥 is
𝑥+2 7𝑥−3 𝑥−3 .
What are all the zeros of 7𝑥 3 −10 𝑥 2 −39𝑥+18 ?
Zero’s are -2, , and 3
Factor theorem---A polynomial f(x) has a factor of (x – k), if and only if f(k) = 0 .

5. A company’s profit C (in thousands of dollars) can be modeled by 𝐶=− 5𝑥 3 +6 𝑥 2 +15𝑥, where x is the number of items produced in thousands. The profit is $14,000 for producing 2000 items. What other number of items would produce about the same profit?
When C = 14, there is a zero of 2. 0= −5𝑥 3 + 6𝑥 2 +15𝑥−14
-10 -8 14 -5 -4 7
The expression that is left, −5 𝑥 2 −4𝑥+7, does not factor, so use the quadratic formula to find an approximate zero that seems appropriate.
Approx , or 850 items!