Fundamental Theorem of Algebra L3

Front

What is the Fundamental Theorem of Algebra?

Back

The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n complex roots, counting multiplicities.

Front

Define end behavior of a polynomial function.

Back

End behavior describes the behavior of the graph of a polynomial function as x approaches positive or negative infinity.

Front

What does it mean if a polynomial function has even degree and positive leading coefficient?

Back

The graph will rise to positive infinity on both ends.

Front

What does it mean if a polynomial function has odd degree and negative leading coefficient?

Back

The graph will fall to negative infinity on the left and rise to positive infinity on the right.

Front

How do you determine the linear factors of a polynomial from its graph?

Back

Identify the x-intercepts of the graph; each x-intercept corresponds to a linear factor of the form (x - r), where r is the x-intercept.

Front

What is a zero of a polynomial function?

Back

A zero of a polynomial function is a value of x that makes the function equal to zero.

Front

How can you find all zeros of a polynomial function from its graph?

Back

By identifying all the points where the graph intersects the x-axis.

Front

What is the significance of the multiplicity of a zero?

Back

The multiplicity of a zero indicates how many times that zero is repeated as a root of the polynomial; it affects the shape of the graph at that intercept.

Front

What is the relationship between the degree of a polynomial and the number of zeros it can have?

Back

A polynomial of degree n can have up to n zeros, including complex and repeated roots.

Front

What does a double root indicate about the graph at that point?

Back

A double root indicates that the graph touches the x-axis at that point but does not cross it.