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Polynomial Functions

Upper and Lower Bound

If we are told that a piece of wood is 12cm long to the nearest cm, then what is the range of possible lengths it could be? It must be at least 11.5cm long to round up to 12cm. But it must be less than 12.5cm, to round down to 12cm.

Rational Zeroes of Polynomials

If P(x) is a polynomial with integer coefficients and if is a zero of P(x) ( P( ) = 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) . We can use the Rational Zeros Theorem to find all the rational zeros of a polynomial.

Leading Coefficient Test and it's behavior

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function f(x)=−x3+5x . ... Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right as shown in the figure.

The Intercepts

Intercepts of lines review (x-intercepts and y-intercepts) The x-intercept is where a line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.

The Turning Points

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.

Dividing Polynomials

Divide the first term of the numerator by the first term of the denominator, and put that in the answer. Multiply the denominator by that answer, put that below the numerator. Subtract to create a new polynomial.

A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of x. We can give a general defintion of a polynomial, and define its degree

Multiplicity of a Root

How many times a particular number is a zero for a given polynomial. 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.