Zeros, Multiplicity, and Graph Shape

Explore zeros as factors, even multiplicity, odd multiplicity.

Lesson 4 of 53 parts Practice Flashcards

1/3

Zeros as Factors

Objective: Explain zeros as factors and use it to solve or interpret a problem.

A real zero r corresponds to factor (x−r) for a polynomial. Substitute r or divide by x−r to verify the claim. Evaluate the polynomial at that input.

Worked examples

Try itZeros, Multiplicity, and Graph Shape — practice 1.1: For p(x)=(x−5)(x+4), find p(5).Show answer

Step 1: The factor x−5 is zero at x=5. Step 2: The product is 0·(9). Step 3: Thus p(5)=0.

Swipe sideways to move between sections

Zeros as Factors

Objective: Explain zeros as factors and use it to solve or interpret a problem.

A real zero r corresponds to factor (x−r) for a polynomial. Substitute r or divide by x−r to verify the claim. Evaluate the polynomial at that input.

Worked examples

Try itZeros, Multiplicity, and Graph Shape — practice 1.1: For p(x)=(x−5)(x+4), find p(5).Show answer

Step 1: The factor x−5 is zero at x=5. Step 2: The product is 0·(9). Step 3: Thus p(5)=0.

Even Multiplicity

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

An even-multiplicity root typically touches the x-axis and turns, because the factor does not change sign across that root. No, it generally touches and turns locally.

Worked examples

Try itZeros, Multiplicity, and Graph Shape — practice 2.1: For p(x)=(x−2)²(x+8), find the multiplicity of x=2.Show answer

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

Odd Multiplicity

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

An odd-multiplicity real root crosses the axis locally; a greater multiplicity may make the crossing flatter. Multiplicity counts repeated factors. The exponent on the associated linear factor.

Worked examples

Try itZeros, Multiplicity, and Graph Shape — practice 3.1: For p(x)=(x−3)²(x+11), find the multiplicity of x=3.Show answer

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

You finished the reading!

Lock it in with a few practice questions, then move on to the next lesson.

Practice Next lesson

Previous lessonPolynomial Functions and End Behavior Next lessonBuilding Polynomial Models