Factoring and Solving Polynomials

Find one root, then shrink the problem.

12 problems · about 22 minutes · A-APR.B.2, A-APR.B.3

NameDate

Figure — use these to answer the problems

The real roots are the crossings. Count them, then remember the rest are complex.

y = 1x³ − 4x + 0

b-4

Figure 1

Straightforward practice. Get the method working first.

5 problems

x³ − 9x = 0. How many real roots?
Answer
2
What does the Rational Root Theorem actually provide?
- A finite list of candidates worth testing.
- The roots themselves.
x³ − x = 0. How many real roots?
Answer
x³ − 8 = 0. What is the real root?
Answer
f(3) = 0. Which factor does that give? Enter 3 for (x − 3).
Answer

The same ideas with more to keep track of.

3 problems

After dividing a cubic by a linear factor, what degree remains?
Answer
A cubic has roots 2, 3i and one more. Imaginary coefficient of the third?
Answer
x³ − 6x² + 11x − 6 factors as (x−1)(x−2)(x−3). Largest root?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Put the solving process in order.
Write 1 to 4 in the boxes to put these in order.
- List the candidate rational roots.
- Test candidates until one gives zero.
- Divide the polynomial by that factor.
- Solve the smaller polynomial that remains.
x³ − 4x = 0. How many real roots?
Answer
The Candidates
For x³ + 2x² − 5x − 6, how many candidate rational roots does the theorem list, counting both signs?
Answer
The Test
f(2) = 0. Is (x − 2) a factor? 1 yes, 0 no.
Answer