Identifying graph based on roots | Polynomial and rational functions | Algebra II | Khan Academy

TL;DR

Use the real zeros of a polynomial function to identify possible graphs by evaluating the function at those points.

Transcript

Use the real 0's of the polynomial function y equal to x to the third plus 3x squared plus x plus 3 to determine which of the following could be its graph. So there's several ways of trying to approach it. One, we could just look at what the 0's of these graphs are or what they appear to be and then see if this function is actually 0 when x is equa... Read More

Key Insights

• 📈 Evaluating a polynomial function at its real zeros helps identify the correct graph.
• 0️⃣ Graphs with zeros that do not align with the real zeros of the function do not accurately represent the polynomial.
• 📈 The number of real and complex zeros can be used to determine if a graph is correct.

Questions & Answers

Q: How can we determine the graph of a polynomial function using its real zeros?

By substituting the real zeros into the function and checking if the result is zero, we can confirm which graph accurately represents the polynomial function.

Q: What can we infer if a polynomial function has three real zeros?

If a polynomial function has three real zeros, it can have no complex zeros. Thus, any graph with two complex zeros or one real and two complex zeros does not represent the given function.

Q: Is it necessary to factorize a polynomial function to find its real zeros?

No, it is not necessary to factorize the polynomial function. However, factoring can provide additional insights into the zeros and help determine the graph.

Q: How do we identify complex zeros of a polynomial function?

Complex zeros occur when the equation x^2 is equal to a negative number, as there are no real numbers with a square equal to a negative. Complex zeros indicate the need for a pair of complex roots.

Summary & Key Takeaways

• Given a polynomial function, we can determine the possible graph by evaluating the function at its real zeros.

• By substituting the real zeros into the function and checking if the result is zero, we can confirm whether a graph is accurate.

• The number of real and complex zeros can help identify the correct graph.