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

What this lesson teaches

The student solves higher-degree polynomial equations using the Rational Root Theorem and factoring.

Straightforward practice. Get the method working first.

5 problems

x³ − 4x = 0. How many real roots?
Answer 3
Why 3.
Why does every odd-degree polynomial have at least one real root?
Answer Complex roots come in pairs, so an odd count cannot be all complex.
Why Pairing off the complex roots always leaves at least one real root behind.
How many roots does a degree-5 polynomial have, counting repeats and complex ones?
Answer 5
Why The degree tells you.
x³ − x = 0. How many real roots?
Answer 3
Why Factor as x(x−1)(x+1).
(x − 4)²(x + 1) = 0. What is the multiplicity of the root 4?
Answer 2
Why Read the exponent.

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 -3
Why Complex roots pair with their conjugates.
x³ − 8 = 0. What is the real root?
Answer 2
Why The cube root of 8.
A quartic has three real roots, one of them repeated once more. How many complex roots does it have?
Answer 0
Why Four roots in total are already accounted for.

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

4 problems

Put 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 before you find one.
Which are candidate rational roots of x³ + 2x² − 5x − 6?
Answer 2; −3
Why Only factors of 6 qualify.
The Candidates: For x³ + 2x² − 5x − 6 = 0, how many candidate rational roots does the theorem list, counting both signs?
Answer 8
Why 8.
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 3
Why 3.