3.4 Real Zeros of Polynomials
Ideas in this lesson
- List and interpret rational-zero candidates
- Find integer zeros by division and repeated factoring
- Find rational zeros with nonmonic factors
- Find irrational real zeros after rational reduction
- Sketch from zeros, multiplicities, and end behavior
- Count possible real zeros using Descartes’ rule
- Certify and search for upper and lower bounds
- Prove absence of rational zeros without excluding real zeros
- Approximate every real zero with a justified viewing interval
- Model cylinder, hemisphere, and cone volumes
- Model a rectangle using its diagonal and area
- Interpret a time-dependent polynomial model and thresholds
- Find every feasible open-box design
- Model a square-base box and space diagonal
- Solve length-plus-girth volume constraints
- Construct examples and prove impossible real-zero patterns
- Derive and apply the depressed-cubic substitution
- Use Cardano’s expression and compare algebraic methods
- Prove the upper and lower bounds theorem
- Prove exact rational and irrational root counts
- Prove rational-zero and division implications and refute false bounds
- Read supplied graphs, then verify candidate zeros exactly
- Find all intersections of two polynomial graphs
- Use deflation and Descartes’ rule together
List and interpret rational-zero candidates · Worked example
List the distinct possible rational zeros of \(4 x^{3} + 3 x^{2} - 5 x + 10\). Explain why a candidate list is not a list of actual zeros. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: \(-10\), \(-5\), \(- \dfrac{5}{2}\), \(-2\), \(- \dfrac{5}{4}\), \(-1\), \(- \dfrac{1}{2}\), \(- \dfrac{1}{4}\), \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), \(1\), \(\dfrac{5}{4}\), \(2\), \(\dfrac{5}{2}\), \(5\), \(10\)
The numerator divides \(10\) and the denominator divides \(4\).
Reduce every signed ratio and remove duplicates: \(-10\), \(-5\), \(- \dfrac{5}{2}\), \(-2\), \(- \dfrac{5}{4}\), \(-1\), \(- \dfrac{1}{2}\), \(- \dfrac{1}{4}\), \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), \(1\), \(\dfrac{5}{4}\), \(2\), \(\dfrac{5}{2}\), \(5\), \(10\).
The theorem gives necessary candidates only; substitute or divide to test them.
Written-work review:
Reduce every signed ratio and remove duplicates: \(-10\), \(-5\), \(- \dfrac{5}{2}\), \(-2\), \(- \dfrac{5}{4}\), \(-1\), \(- \dfrac{1}{2}\), \(- \dfrac{1}{4}\), \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), \(1\), \(\dfrac{5}{4}\), \(2\), \(\dfrac{5}{2}\), \(5\), \(10\).
The theorem gives necessary candidates only; substitute or divide to test them.
Written-work review:
- The numerator divides \(10\) and the denominator divides \(4\).
- Reduce every signed ratio and remove duplicates: \(-10\), \(-5\), \(- \dfrac{5}{2}\), \(-2\), \(- \dfrac{5}{4}\), \(-1\), \(- \dfrac{1}{2}\), \(- \dfrac{1}{4}\), \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), \(1\), \(\dfrac{5}{4}\), \(2\), \(\dfrac{5}{2}\), \(5\), \(10\).
- The theorem gives necessary candidates only; substitute or divide to test them.