The theorems and rules I'm asking students to apply are:
- Intermediate Value Theorem
- Remainder Theorem
- Factor Theorem
- The Rational Zero Test
- Descarte's Rule of Signs
- Upper Bound and Lower Bound Rules
We use Pre-Calculus with Limits: A Graphing Approach by Larson, Hostetler, and Edwards.
Some comments that I'm looking for:
- p/q is work for the Rational Zero Test. It is used to find all the possible rational zeros.
- f(x)=2x3+5x2-37x-60 has 1 variation in sign (sign change in the coefficients) so there's 1 positive real zero by Descarte's Rule of Signs. (Work is not shown)
- f(-x)=-2x3+5x2+37x-60 has 2 variations in sign (sign change in the coefficients) so there's 2 or 0 negative real zeros by Descarte's Rule of Signs. (Work is not shown)
- 5 is an upper bound because the last row is either positive or zeros and 5 is positive. This means no real zeros above 5.
- 6 should be skipped since 5 is an upper bound. However, this may still be useful if we're graphing the function. The remainder of 330 means f(6)=330. So we have as point (6,330).
- 3 has a remainder of -72. Since f(3)=-72 and f(5)=180, and since f(x) is a polynomial (continuous everywhere), there must be a zero in the interval [3,5] by Intermediate Value Theorem. 4 is a good number to try.
- 2 is unnecessary since we know there is only 1 positive zero and it is between 3 and 5.
- 4 happens to be a zero. The quotient is 2x2+13x+15. Student should attempt to factor this quadratic to save time.
- 1 is unnecessary since there is only 1 positive real zero by Descarte's Rule of Signs and it's been found that 4 is that zero.
- -6 is a lower bound since the last row is alternately positive and negative and -6 is negative. This means no real zeroes below -6. The original f(x) should not have been used here. The quotient of 2x2+13x+15 should have been used instead, if factoring is not immediately obvious.
- -10 should not have been tried if all we are doing is factoring. We already know -6 is a lower bound.
- -5 is a zero.
- 4 is used here to find out the last factor. We found 4 was a zero from an earlier step.
- The polynomial f(x)=2x3+5x2-37x-60 was factored to f(x)=(x-4)(x+5)(2x+3)
It seems to me that, for many students, time is better spent practicing how to consistently and accurately divide using synthetic division. Here's some questions to math teachers:
- Do you teach these theorems and rules?
- If you don't cover them, why not?
- If you do, how do you cover them (how much time/depth)?
- How do you teach/motivate learning these theorems and rules?
I'd like to hear your thoughts.