Polynomial Function

The graphical concept of x- and y-intercepts is pretty simple. The x-intercepts are where the graph crosses the x-axis, and the y-intercepts are where the graph crosses the y-axis. The problems start when we try to deal with intercepts algebraically.

Find the x- and y-intercepts of 25x2 + 4y2 = 9
Using the definitions of the intercepts, I will proceed as follows:

x-intercept(s):

y = 0 for the x-intercept(s), so:

25x2 + 4y2 = 9
25x2 + 4(0)2 = 9
25x2 + 0 = 9
x2 = 9/25
x = ± ( 3/5 )

Then the x-intercepts are the points ( 3/5, 0) and ( –3/5, 0) y-intercept(s):
x = 0 for the y-intercept(s), so:
25x2 + 4y2 = 9

25(0)2 + 4y2 = 9

0 + 4y2 = 9

y2 = 9/4

y = ± ( 3/2 )

Then the y-intercepts are the points (0, 3/2 ) and (0, –3/2 )

Upper Bound Theorem: If you divide a polynomial function f(x) by (x - c), where c > 0, using synthetic division and this yields all non-negative numbers, then c is an upper bound to the real roots of the equation f(x) = 0. Using synthetic division, I found the integral upper bound to be 2. All the coefficients and the remainder are non-negative.

Now on to the lower bound.
A lower bound is an integer less than or equal to the least real zero.

Lower Bound Theorem:
If you divide a polynomial function f(x) by (x - c), where c < 0, using synthetic division and this yields alternating signs, then c is a lower bound to the real roots of the equation f(x) = 0. Special note that zeros can be either positive or negative.
Using synthetic division again, I tested -1, -2, -3, -4, and -5.
Only -5 yielded alternating signs in the coefficients and remainder of the quotient.
This says that -5 is a lower bound and all the real zeros of can be found in the interval
Now, -4 failed the lower bound theorem test because the quotient did not produce alternating signs. This would suggest that -4 is not a lower bound, right?
However, upon further inspection, the actual zeros are {-3.62, -1.38, 2)
This would seem to indicate that a lower bound (in fact the greatest lower bound) should have been -4.

For example, in the polynomial function f(x) = (x – 3)4(x – 5)(x – 8)2, the zero 3 has multiplicity 4, 5 has multiplicity 1, and 8 has multiplicity 2. Although this polynomial has only three zeros, we say that it has seven zeros counting multiplicity. See also. Double root, triple root, fundamental theorem of algebra.

A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. For instance, the quadratic (x + 3)(x – 2) has the zeroes x = –3 and x = 2, each occuring once.

A turning point is a point at which the derivative changes sign. A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points.

To find the y-intercept, put x = 0 into the equation and work out the y-coordinate. To find the x-coordinate, put y = 0 in the equation and solve the quadratic equation to get the x-coordinates. To find the vertex (turning point), add the x-intercepts together and divide by 2.

polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division.

This means that:
7x2+x−8x−1=7x+8

The Rational Zeros Theorem states:
If P(x) is a polynomial with integer coefficients and if p/q is a zero of P(x) ( P(p/q) = 0 ), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x) .

The graph of the polynomial function f(x)=anxn+an−1xn−1+...+a1x+a0f(x)=anxn+an−1xn−1+...+a1x+a0 eventually rises or falls depends on the leading coefficient (an)(an) and the degree of the polynomial function.