Polynomial Function
Turning points of polynomial functions
A turning point of a function is a point where the graph of the function changes from sloping
downwards to sloping upwards, or vice versa. So the gradient changes from negative to positive,
or from positive to negative. Generally speaking, curves of degree n can have up to (n − 1)
turning points.
Rational root theorem
The Rational Zeros Theorem. The Rational Zeros Theorem states: 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) .
he rational zeros theorem (also called the rational root theorem) is used to check whether a polynomial has rational roots (zeros). It provides a list of all possible rational roots of the polynomial equation , where all coefficients are integers.
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.
Graphing Polynomial Function
1. Determine all the zeroes of the polynomial and their multiplicity. ...
2. Determine the y-intercept, .
3. Use the leading coefficient test to determine the behavior of the polynomial at the end of the graph.
4. Plot a few more points.
The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
To find the x-intercept of a given linear equation, plug in 0 for 'y' and solve for 'x'. To find the y-intercept, plug 0 in for 'x' and solve for 'y'. In this tutorial, you'll see how to find the x-intercept and the y-intercept for a given linear equation.
Multiplicity of a root, 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.
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.
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.
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.
Factor theorem
The factor theorem is a theorem linking factors and zeros of a polynomial. It is commonly applied to factorizing and finding the roots of polynomial equations. The theorem states that isa factor of a polynomial f(x)if ;that is, r is a root of f(x). ... If , then the remainder is 0 and , showing that is a factor of f(x).
What is a polynomial?
A polynomial of degree n is a function of the form
f(x) = anx
n + an−1x
n−1 + . . . + a2x
2 + a1x + a0
where the a’s are real numbers (sometimes called the coefficients of the polynomial). Although
this general formula might look quite complicated, particular examples are much simpler. For
example,
f(x) = 4x
3 − 3x
2 + 2
is a polynomial of degree 3, as 3 is the highest power of x in the formula. This is called a cubic
polynomial, or just a cubic. And
f(x) = x
7 − 4x
5 + 1
is a polynomial of degree 7, as 7 is the highest power of x. Notice here that we don’t need every
power of x up to 7: we need to know only the highest power of x to find out the degree. An
example of a kind you may be familiar with is
f(x) = 4x
2 − 2x − 4
which is a polynomial of degree 2, as 2 is the highest power of x. This is called a quadratic.
Functions containing other operations, such as square roots, are not polynomials. For example,
f(x) = 4x
3 +
√
x − 1
is not a polynomial as it contains a square root. And
f(x) = 5x
4 − 2x
2 + 3/x
is not a polynomial as it contains a ‘divide by x’.
Dividing polynomials using synthetic method
In order to divide polynomials using synthetic division, you must be dividing by a linear expression and the leading coefficient (first number) must be a 1. For example, you can use synthetic division to divide by x + 3 or x – 6, but you cannot use synthetic division to divide by x2 + 2 or 3x2 – x + 7.
Remainder theorem
In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial by a linear polynomial is equal to. In particular, is a divisor of if and only if.