Building Polynomial Models

Explore constructing from roots, using a point, checking a model.

Lesson 5 of 53 parts Practice Flashcards

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Constructing from Roots

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

Given roots r₁ and r₂, a simplest quadratic has form k(x−r₁)(x−r₂). More roots add factors, and multiplicities must be specified when relevant. Roots alone do not fix every y-value.

Worked examples

Try itBuilding Polynomial Models — practice 1.1: A polynomial with roots 2 and 5 is p(x)=k(x−2)(x−5). If p(0)=20, find k.Show answer

Step 1: Set x=0: p(0)=k(−2)(−5)=10k. Step 2: Solve 10k=20. Step 3: Divide to obtain k=2.

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Constructing from Roots

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

Given roots r₁ and r₂, a simplest quadratic has form k(x−r₁)(x−r₂). More roots add factors, and multiplicities must be specified when relevant. Roots alone do not fix every y-value.

Worked examples

Try itBuilding Polynomial Models — practice 1.1: A polynomial with roots 2 and 5 is p(x)=k(x−2)(x−5). If p(0)=20, find k.Show answer

Step 1: Set x=0: p(0)=k(−2)(−5)=10k. Step 2: Solve 10k=20. Step 3: Divide to obtain k=2.

Using a Point

Objective: Explain using a point and use it to solve or interpret a problem.

Substitute a known point into the product form to find its nonzero scale factor, then verify the completed rule against all given conditions. It may not determine the scale factor.

Worked examples

Try itBuilding Polynomial Models — practice 2.1: A polynomial with roots 3 and 8 is p(x)=k(x−3)(x−8). If p(0)=48, find k.Show answer

Step 1: Set x=0: p(0)=k(−3)(−8)=24k. Step 2: Solve 24k=48. Step 3: Divide to obtain k=2.

Checking a Model

Objective: Explain checking a model and use it to solve or interpret a problem.

A fitted polynomial can match selected points yet behave poorly elsewhere. State a meaningful domain and compare residuals or additional observations. The fitted behavior may not persist outside observed values.

Worked examples

Try itBuilding Polynomial Models — practice 3.1: For p(x)=x³−4x, evaluate p(11).Show answer

Step 1: Substitute 11: p(11)=11³−4(11). Step 2: Compute 1331−44. Step 3: The output is 1287.

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