Solving Polynomial Equations

Explore real roots, nonreal roots, multiplicity and checks.

Lesson 5 of 53 parts Practice Flashcards

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Real Roots

Objective: Explain real roots and use it to solve or interpret a problem.

Use factors or graph intercepts to find real roots and substitute candidates into the original polynomial. A real root of the polynomial equation.

Worked examples

Try itSolving Polynomial Equations — practice 1.1: For p(x)=(x−5)(x+2), find p(5).Show answer

Step 1: The factor x−5 is zero at x=5. Step 2: The product is 0·(7). Step 3: Thus p(5)=0.

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Real Roots

Objective: Explain real roots and use it to solve or interpret a problem.

Use factors or graph intercepts to find real roots and substitute candidates into the original polynomial. A real root of the polynomial equation.

Worked examples

Try itSolving Polynomial Equations — practice 1.1: For p(x)=(x−5)(x+2), find p(5).Show answer

Step 1: The factor x−5 is zero at x=5. Step 2: The product is 0·(7). Step 3: Thus p(5)=0.

Nonreal Roots

Objective: Explain nonreal roots and use it to solve or interpret a problem.

A quadratic with negative discriminant has two nonreal complex roots; its real graph does not meet the x-axis. State whether solutions are requested over reals or complex numbers. No, nonconstant polynomials have complex roots counting multiplicity.

Worked examples

Try itSolving Polynomial Equations — practice 2.1: For x²+9=0, how many real solutions exist?Show answer

Step 1: Rewrite as x²=−9. Step 2: A real square cannot be negative. Step 3: Therefore there are zero real solutions.

Multiplicity and Checks

Objective: Explain multiplicity and checks and use it to solve or interpret a problem.

Repeated factors count repeated roots; verify algebraic factorization and candidate values. A numerical approximation needs its precision stated. Division or algebraic manipulation can introduce errors.

Worked examples

Try itSolving Polynomial Equations — practice 3.1: For p(x)=(x−4)²(x+11), find the multiplicity of x=4.Show answer

Step 1: The factor x−4 is squared. Step 2: The root x=4 appears twice. Step 3: Its multiplicity is 2.

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