Cogito
Integrated Math 3 · Chapter 1 · Lesson 2
Factoring and Solving Polynomials
Find one root, then shrink the problem.
12 problems · about 22 minutes · A-APR.B.2, A-APR.B.3
What this lesson teaches
The student solves polynomial equations using the Rational Root Theorem and division.
- A root and a factor are the same fact, by the Factor Theorem.
- The Rational Root Theorem gives a finite list of candidates.
- Divide out each root found and solve what remains.
Warm Up
Straightforward practice. Get the method working first.
5 problemsx³ − 9x = 0. How many real roots?
Answer 3
Why 3.
What does the Rational Root Theorem actually provide?
Answer A finite list of candidates worth testing.
Why Candidates, not answers.
x³ − x = 0. How many real roots?
Answer 3
Why x(x−1)(x+1).
x³ − 8 = 0. What is the real root?
Answer 2
Why The cube root of 8.
f(3) = 0. Which factor does that give? Enter 3 for (x − 3).
Answer 3
Why The Factor Theorem.
Build It Up
The same ideas with more to keep track of.
3 problemsAfter dividing a cubic by a linear factor, what degree remains?
Answer 2
Why One lower.
A cubic has roots 2, 3i and one more. Imaginary coefficient of the third?
Answer -3
Why Conjugate pair.
x³ − 6x² + 11x − 6 factors as (x−1)(x−2)(x−3). Largest root?
Answer 3
Why Read the factors.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the solving process in order.
Answer 1. List the candidate rational roots. 2. Test candidates until one gives zero. 3. Divide the polynomial by that factor. 4. Solve the smaller polynomial that remains.
Why You cannot divide by a factor you have not found.
x³ − 4x = 0. How many real roots?
Answer 3
Why x(x−2)(x+2).
The Candidates: For x³ + 2x² − 5x − 6, how many candidate rational roots does the theorem list, counting both signs?
Answer 8
Why 8.
The Test: f(2) = 0. Is (x − 2) a factor? 1 yes, 0 no.
Answer 1
Why Yes.