Multiplicity of zeros of polynomials | Polynomial graphs | Algebra 2 | Khan Academy

TL;DR

Polynomials can have multiple zeros with different multiplicities, causing different behaviors in their graphs.

Transcript

• [Instructor] So what we have here are two different polynomials, P1 and P2. And they have been expressed in factored form and you can also see their graphs. This is the graph of Y is equal to P1 of x in blue, and the graph of Y is equal to P2 x in white. What we're going to do in this video is continue our study of zeros, but we're gonna look at ... Read More

Key Insights

• 📈 Polynomials can have zeros with different multiplicities, affecting their graph behavior.
• 🫰 Zeros with odd multiplicities cause a sign change when crossing the x-axis, while zeros with even multiplicities do not.
• 🟰 The number of distinct zeros in a polynomial is at most equal to its degree.

Questions & Answers

Q: What is the significance of multiplicity in polynomials?

Multiplicity refers to the number of times a zero appears in the factored form of a polynomial. It affects the behavior of the graph at that zero and determines whether there is a sign change when crossing the x-axis.

Q: How does the graph behave at zeros with odd multiplicity?

When a zero has an odd multiplicity (1, 3, 5, etc.), the graph will cross the x-axis at that zero. This results in a sign change in the function.

Q: What happens at zeros with even multiplicity?

Zeros with even multiplicities (2, 4, 6, etc.) do not cause a sign change in the function. The graph touches or intersects the x-axis at these zeros without crossing it.

Q: How is the number of zeros related to the degree of a polynomial?

The number of zeros in a polynomial is at most equal to its degree. If a zero has a multiplicity higher than one, it contributes fewer distinct zeros to the polynomial.

Summary & Key Takeaways

• Polynomials P1 and P2 are analyzed in terms of their zeros and graph behavior.

• P1 has three distinct zeros, and the graph crosses the x-axis at each zero.

• P2 has one zero with a multiplicity of one and one zero with a multiplicity of two, causing different graph behavior at each zero.