Calculus Path
Statistics Path
MTH111
College Algebra
PreCalculus, part 1
vs.
MTH105
Math in Society
Math for "Real" Life
Which class is right for you?
Is calculus in your future or NOT?
§ 3.2 Polynomial Factors, Zeros, and Remainders
Is `x-2` a factor of `x^3-3x^2+4x-5`
One way to check is with long division to find the quotient and remainder. If the remainder is zero then `x-2` is a factor.
`x-2 \ bar(")"\ x^3-3x^2+4x-5)`
What do you multiply by `x` to get `x^3`?
Guess: `x^2` and multiply:
| `x^2` | ||||
| `x-2` | `)x^3` | `-3x^2` | `+4x` | `-5` |
| `x^3` | `-2x^2` |
Now subtract:
| `x^2` | |||||
| `x-2` | `)` | `x^3` | `-3x^2` | `+4x` | `-5` |
| `-(x^3` | `-2x^2)` | ||||
| `-1x^2` | `+4x` |
What would you multiply by `x` to get `-1x^2`?
Guess: `-1x` and multiply.
| `x^2` | `-1x` | ||||
| `x-2` | `)` | `x^3` | `-3x^2` | `+4x` | `-5` |
| `-(x^3` | `-2x^2)` | ||||
| `-1x^2` | `+4x` | ||||
| `-1x^2` | `+2x` |
Now subtract:
| `x^2` | `-1x` | ||||
| `x-2` | `)` | `x^3` | `-3x^2` | `+4x` | `-5` |
| `-(x^3` | `-2x^2)` | ||||
| `-1x^2` | `+4x` | ||||
| `-(` | `-1x^2` | `+2x)` | |||
| `2x` | `-5` |
What would you multiply by `x` to get `2x`?
Guess: `2` and multiply.
| `x^2` | `-1x` | `+2` | |||
| `x-2` | `)` | `x^3` | `-3x^2` | `+4x` | `-5` |
| `-(x^3` | `-2x^2)` | ||||
| `-1x^2` | `+4x` | ||||
| `-(` | `-1x^2` | `+2x)` | |||
| `2x` | `-5` | ||||
| `2x` | `-4` |
Now subtract.
| `x^2` | `-1x` | `+2` | |||
| `x-2` | `)` | `x^3` | `-3x^2` | `+4x` | `-5` |
| `-(x^3` | `-2x^2)` | ||||
| `-1x^2` | `+4x` | ||||
| `-(` | `-1x^2` | `+2x)` | |||
| `2x` | `-5` | ||||
| `-(` | `2x` | `-4)` | |||
| `-1` |
Solution: Since the remainder is `-1` and not `0` then `x-2` is not a factor of `x^3-3x^2+4x-5`
CHECK:
If `x-2` is a factor then `f(2)=0`
`f(2)=(2)^3-3(2)^2+4(2)-5=-1` Not `0`.
Chapter 4: Finance
Problem Situation: Find the monthly payment for a home mortgage of $148,500 at 4% for 30 years.
Here is a spreadsheet for this loan.
Details:
| House = | $165,000.00 |
| 10% down = | $16,500.00 |
| Loan = | $148,500.00 |
| APR = | 4% |
| Years = N = | 30 |
| k = | 12 |
| Payment = | $708.96 |
| Total Payments = | $255,226.22 |
| Total Interest = | $106,726.22 |
The first few months:
| months | Payment | Interest Due | Balance Reduction | Balance |
| 0 | $148,500.00 | |||
| 1 | $708.96 | $495.00 | $213.96 | $148,286.04 |
| 2 | $708.96 | $494.29 | $214.67 | $148,071.36 |
| 3 | $708.96 | $493.57 | $215.39 | $147,855.97 |
| 4 | $708.96 | $492.85 | $216.11 | $147,639.86 |
| 5 | $708.96 | $492.13 | $216.83 | $147,423.04 |
| 6 | $708.96 | $491.41 | $217.55 | $147,205.48 |
Disclaimer:
You should check with your advisor before taking a math class leading to calculus. Algebra classes are known to cause stress, anger, depression, and occasionally rashes, as well as lead to calculus. Please consult a physician if negative symptoms persist.