Recall that a polynomial function has the form...what does this tell you about the number of x-intercepts of the graph of f (x)? Explain.

Question

Recall that a polynomial function has the form where n is a positive integer and . 

Consider a polynomial of even degree.

If  and , what does this tell you about the number of x-intercepts of the graph of ? Explain.

Note:

Please write neatly and clearly.

Grade 12 Advanced Function.

Answer 1

Although it requires complex analysis to prove it, every polynomial of degree n with real coefficients has exactly n roots, some of which may be complex. Every real root corresponds to an x intercept.

For a₀ >0, the y intercept is also greater than 0 and equal to a₀.

Since a_n < 0, the graph goes to -∞ when |x| is large.

This means the graph must cross the x axis at least twice, i.e. the polynomial has at least 2 real roots.

This last comment depends on the fact that polynomials are continuous on the whole real line.