Solving Polynomial Equations

Hunt the rational roots, then factor what is left.

12 problems · about 23 minutes · A-APR.B.3, N-CN.C.9

NameDate

Figure — use these to answer the problems

Move the coefficients and count how many times the curve meets the x-axis.

y = 1x³ − 3x + 0

b-3

Figure 1

Straightforward practice. Get the method working first.

5 problems

x³ − 4x = 0. How many real roots?
Answer
2
Why does every odd-degree polynomial have at least one real root?
- Complex roots come in pairs, so an odd count cannot be all complex.
- Because odd numbers are larger.
How many roots does a degree-5 polynomial have, counting repeats and complex ones?
Answer
x³ − x = 0. How many real roots?
Answer
(x − 4)²(x + 1) = 0. What is the multiplicity of the root 4?
Answer

The same ideas with more to keep track of.

3 problems

A cubic has roots 2, 3i and one more. What is the third root, as a coefficient of i?
Answer
x³ − 8 = 0. What is the real root?
Answer
A quartic has three real roots, one of them repeated once more. How many complex roots does it have?
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.
10
Which are candidate rational roots of x³ + 2x² − 5x − 6?
- 2
- −3
- 4
- 5
- Tick every box that applies.
The Candidates
For x³ + 2x² − 5x − 6 = 0, how many candidate rational roots does the theorem list, counting both signs?
Answer
The Box
A box has volume x³ − 6x² + 11x − 6 cubic units, which factors as (x−1)(x−2)(x−3). What is the largest root?
Answer