Polynomial Functions and End Behavior

Explore degree and leading term, even and odd degree, interpreting graphs.

Lesson 3 of 53 parts Practice Flashcards

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Degree and Leading Term

Objective: Explain degree and leading term and use it to solve or interpret a problem.

The highest nonzero power and its coefficient dominate polynomial behavior for large |x|. The degree does not depend on smaller terms. Their relative influence shrinks as |x| grows.

Worked examples

Try itPolynomial Functions and End Behavior — practice 1.1: What is the degree of 3x^5+7x²+1 (c>2)?Show answer

Step 1: Identify the greatest nonzero exponent. Step 2: The x^5 term has coefficient 3≠0. Step 3: The degree is 5.

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Degree and Leading Term

Objective: Explain degree and leading term and use it to solve or interpret a problem.

The highest nonzero power and its coefficient dominate polynomial behavior for large |x|. The degree does not depend on smaller terms. Their relative influence shrinks as |x| grows.

Worked examples

Try itPolynomial Functions and End Behavior — practice 1.1: What is the degree of 3x^5+7x²+1 (c>2)?Show answer

Step 1: Identify the greatest nonzero exponent. Step 2: The x^5 term has coefficient 3≠0. Step 3: The degree is 5.

Even and Odd Degree

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

A polynomial of even degree has ends heading the same way; an odd-degree polynomial has ends heading opposite ways, with direction set by its leading sign. No, the leading term controls them.

Worked examples

Try itPolynomial Functions and End Behavior — practice 2.1: For p(x)=−4x^17+3, is its right-end output positive or negative? Answer 1 for positive, −1 for negative.Show answer

Step 1: The degree 17 is odd. Step 2: The leading coefficient −4 is negative. Step 3: As x→+∞, the output tends to −∞; choose −1.

Interpreting Graphs

Objective: Explain interpreting graphs and use it to solve or interpret a problem.

A polynomial graph is continuous and smooth in the usual classroom model; degree bounds the number of real zeros and turning points. No, it has at most four distinct zeros.

Worked examples

Try itPolynomial Functions and End Behavior — practice 3.1: What is the degree of 2x^11+4x²+1 (c>2)?Show answer

Step 1: Identify the greatest nonzero exponent. Step 2: The x^11 term has coefficient 2≠0. Step 3: The degree is 11.

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