Polynomial Behavior

Question

1) A polynomial starts with the term -3x^8. Tell me about each of these characteristics:

1. degree:
2. # of local maxima/minima possible:
3. end behavior:
4. # of zeros possible:

Answer 1

1. The degree of a polynomial is given by the largest exponent (degree) of any monomial within the polynomial. In this case, the largest exponent is 8, so that's the degree of the polynomial.
2. The total possible maxima/minima (also referred to sometimes as "bumps") is always n-1 (where n is the degree). So in this case n-1=7.
3. End behavior is determined by the even or odd status of the degree of the polynomial and the sign of the leading coefficient. The degree is 8, which is even, so we know that the arms of the graph will point the same way. Since the leading coefficient is negative, we know that both arms will be pointing down. Usually, you will describe this with a notation indicating that as x approaches ∞, y approaches -∞, and when x approaches -∞, y approaches -∞.
4. Finally, the number of possible zeroes is equal to the degree.

Keep in mind that you can remember all this by memorizing just the behavior of a quadratic (parabola) and a cubic equation. All the other functions follow their pattern.

Best wishes!

Answer 2

Hi Jacqueline:

Let's tackle this one!

Our polynomial has highest power of x as 8. so, it's of form f(x) = -3x⁸ + b₁x⁷ + b₂x⁶ + . . . + b₈

where b₁, b₂, . . . . b₈ are constants (some or all of which may be zero.)

1. The degree of a polynomial is just the highest power of x, which is 8.
2. A function, f, has a local maximum/minimum for all values of x where f'(x) = 0

Now, f'(x) = -24x⁷ + 7b₁x⁶ +. . . .

If we set f'(x) = 0, we have a polynomial of degree 7. By the fundamental theorem of algebra, this has exactly 7 roots, some or all of which may be complex numbers. Since f' is an odd function (degree is an odd number, i.e., 7), it must cross the X-axis at least once, meaning it has at least one real root.

Therefore, f'(x) = 0 for at lest 1 and at most 7 real numbers. So, our original polynomial has at least one and at most 7 local maxima and minima.

3.I assume "end behavior" means what happens to f(x) as x -> -∞ and +∞

As x-> +-∞, the dominant term in the polynomial is -3x⁸. Now x⁸ is an even power so it goes to +∞ whether x is positive or negative, i.e., lim {x-> +-∞} (f(x)) is -∞

So f goes off to -∞, on the left and on the right.

1. the function f rises up rom -∞ on the left and it crosses the X-axis 0, 2, 4, 6, or 8 times. These are the number of possible zeros (or roots). The exact number can't be determined without knowing the other terms of the polynomial.

Answer 3

Hi there!

1) The degree is the highest exponent in the polynomial: In this case, 8.

2) In general (without solving for them), a function with degree n, has a maximum of n-1 local max/mins. So the function can have up to 7 local extrema (max's/mins).

3) A function with a positive coefficient goes to +infinity as x gets bigger (goes really far to the right).

A function with a negative coefficient goes to -infinity as x goes really far to the right.

A function with an even degree the function is symmetric, when the degree is odd the function goes the other way (right goes up, left goes, down, etc.)

Our function has a negative coefficient and an even degree, so the function goes to negative infinity on both ends.

4) # of possible zeros = degree of function = 8.

I hope this helps!

Answer 4

1) 8 th

2) 1 global maximum

3) x → +∞, y → -∞; x → -∞, y → -∞

4) 1, x = 0