The polynomial function y = x

^{4}+ 3x^{3}- 9x^{2}- 23x - 12 graphed above, has only**three zeros**, at 'x' = -4, -1and 3. This is one less than the maximum of**four zeros**that a polynomial of degree four can have.Just so, what is the maximum number of zeros in a cubic polynomial?

Regardless of odd or even, any polynomial of positive order can have a maximum number of zeros equal to its order. For example, a cubic function can have as many as

**three**zeros, but no more. This is known as the fundamental theorem of algebra.How do you know how many real roots a polynomial has?

Total Number of Roots. On the page Fundamental Theorem of Algebra we explain that a polynomial will have exactly as many roots as its degree (the degree is the highest exponent of the polynomial). So we know one more thing: the degree is 5 so there are

**5 roots**in total.1

## What is the maximum number of real roots a polynomial of degree 3 can have?

According to the complex conjugate root theorem, the number of complex roots of a polynomial is always equal to its degree. Since odd degree polynomials have a maximum of

**2**turning points, they can have a maximum of 3 real roots.2

## Which polynomial function has an end behavior of up and down?

Since the

**leading**coefficient of this odd-degree polynomial is positive, then its end-behavior is going to mimic that of a positive cubic. Therefore, the end-behavior for this polynomial will be: "Down" on the**left**and "up" on the right.3

## How many zeros can a quadratic function have?

Roots are also called x-intercepts or zeros. A quadratic function is graphically represented by a parabola with vertex located at the origin, below the x-axis, or above the x-axis. Therefore, a quadratic function may have one,

**two**, or zero roots.4

## Can a polynomial of degree n have fewer than n roots under what conditions?

If we count

**roots**according to their multiplicity (see The Factor Theorem),**then**: A**polynomial of degree n can have**only an even number**fewer than n**real**roots**. Thus, when we count multiplicity, a cubic**polynomial can have**only three**roots**or one**root**; a quadratic**polynomial can have**only two**roots**or zero**roots**.5

## What is a rational zero?

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).6

## Can a cubic polynomial have no real zeros?

As you point out, this is obvious when you look at such a

**cubic**equation graphically.**No**, it's not possible. In fact**any polynomial**of odd degree MUST**have**at least 1**real**root (since complex**roots**occur in conjugate pairs).7

## What is the meaning of complex roots?

**Complex Roots**. A given quadratic equation ax

^{2}+ bx + c = 0 in which b

^{2}-4ac < 0 has two

**complex roots**: x = , . Therefore, whenever a

**complex**number is a

**root**of a polynomial with real coefficients, its

**complex**conjugate is also a

**root**of that polynomial.

8

## Can you have a square root in a polynomial function?

This is called a quadratic. Functions containing other operations, such as

**square roots**, are not**polynomials**. For example, f(x)=4x3 + √x − 1 is not a**polynomial**as it contains a**square root**.9

## What is the end behavior of a function?

The end behavior of a polynomial function is the behavior of the graph of as approaches

**positive**infinity or**negative**infinity. The degree and the**leading**coefficient of a polynomial function determine the end behavior of the graph.10

## Can a cubic equation have no real roots?

Just as a quadratic equation may

**have**two**real roots**, so a**cubic**equation**has**possibly three. But unlike a quadratic equation which may**have no real**solution, a**cubic**equation always**has**at least one**real root**. We**will**see why this is the case later.11

## What are the roots of a polynomial?

A

**root of a polynomial**is a number such that . The fundamental theorem of algebra states that a**polynomial**of degree has**roots**, some of which may be degenerate. For example, the**roots**of the**polynomial**. (1) are , 1, and 2.12

## What is the leading coefficient in a graph?

Example: Use the

**Leading Coefficient**Test to determine the end behavior of the**graph**of the polynomial function f ( x ) = − x 3 + 5 x . Solution: 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.13

## What is the fundamental theorem of algebra?

The

**fundamental theorem of algebra**states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero.14

## How does multiplicity affect the graph of a function?

The

**multiplicity**of a root**affects**the shape of the**graph**of a polynomial. Specifically, If a root of a polynomial has odd**multiplicity**, the**graph**will cross the x-axis at the the root. If a root of a polynomial has even**multiplicity**, the**graph**will touch the x-axis at the root but will not cross the x-axis.15

## How do you put a polynomial in standard form?

One way to

**write**a**polynomial**is in**standard form**. In order to**write**any**polynomial in standard form**, you look at the degree of each term. You then**write**each term in order of degree, from highest to lowest, left to**write**. First, look at the degrees for each term in the expression.16

## How do you find multiplicity?

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**.17

## How do you know if the multiplicity is odd or even?

**And the beautiful thing is**

- If the multiplicity is odd, the graph will cross the x-axis at that zero. That is, it will change sides, or be on opposite sides of the x-axis.
- If the multiplicity is even, the graph will touch the x-axis at that zero. That is, it will stay on the same side of the axis.

18

## What is odd and even multiplicity?

The graph crosses the x-axis at roots of

**odd multiplicity**and bounces off (not goes through) the x-axis at roots of**even multiplicity**. A non-zero polynomial function is always non-negative if and only if all its roots have an**even multiplicity**and there exists x_{0}such that f(x_{0}) > 0.19

## What does the multiplicity of a zero mean?

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.20

## How do you find the multiplicity of a zero?

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**.