Polynomial Functions (Week 1-2)

Polynomial Functions (Week 1-2)

by: MARY JOY P. LERDON

Recall...​

What is a POLYNOMIAL EXPRESSION?

Open Ended

How will we know that the given expression is polynomial expression?

A polynomial expression has ______________.

Is the given expression polynomial ?​

Yes/No

Multiple Choice

$$x^{\frac{1}{2}}+5x^3-4$$  

• Yes

• No

Multiple Choice

$$-4x^{\sqrt[]{2}}+15x+3$$  

• Yes

• No

Multiple Choice

$$x^2+5x-4$$  

• Yes

• No

Multiple Choice

$$-4x^2+5x^3+13$$  

• Yes

• No

Multiple Choice

$$\frac{3}{x^2}+5x^{10}-20x+5$$  

• Yes

• No

Multiple Choice

$$-\frac{4}{x^2}+5x^3+13x$$  

• Yes

• No

​Polynomial Function

​A polynomial of degree n is a function of the form P(x)=a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ , where n is a nonnegative integer, the a’s such as, a_n, a_n-1 … , a₂ , a₁, a₀ are real numbers called the coefficients. a_nxⁿ is the leading term, a_n is the leading coefficient and a₀ is the constant term.

The terms of a polynomial may be written in any order. However, if they are written in decreasing powers of x, we say that the polynomial is in standard form.

​Other ways in writing Polynomial Functions

​Other than P(x), a polynomial function can also be denoted by f(x). Sometimes, a polynomial function is represented by a set of P of ordered pairs (x, y). Thus, a polynomial function can be written in different ways, like the following:​

• ​P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀

• ​f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀

• ​y = a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀

​

Polynomials may also be written in factored form and as a product of irreducible factors, that is, a factor that can no longer be factored

Which are polynomial functions?​

Multiple Choice

$$-4x^2+5x^3+13$$  

• Yes

• No

Multiple Choice

$$y=-4x^2+13$$  

• Yes

• No

Multiple Choice

$$5x^3-7x^2+4x+13=-2$$  

• Yes

• No

Multiple Choice

$$P\left(x\right)\ =\ -4x^2+\frac{3}{x}+13$$  

• Yes

• No

Multiple Choice

$$f\left(x\right)=-4x^2+5x+13$$  

• Yes

• No

Type of Polynomial according to degree

​Identify the degree and the type the given polynomial.

Graph of Polnomial Functions​

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​Graph of Polynomial Functions

​1. The graph of any type of function must pass the vertical line test.

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​Graph of Polynomial Functions

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2. Every polynomial function with real coefficients has the set of real numbers as its domain; hence it is continuous function. This means that the graph of a polynomial function has no breaks, holes or gaps.

​Graph of Polynomial Functions

3. A polynomial equation of the nth degree cannot have more than n roots. This only means that the graph cannot intersect the x-axis more than n times.

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​Graph of Polynomial Functions

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4. A graph of a polynomial function has only smooth, rounded turns. A polynomial function cannot have a sharp turn.

Poll

Which of the following is a graph of polynomial function? Check all correct answer/s.

​​Zeros of Polynomial Function

It can be shown that for a polynomial function of degree n, the following statements are true:

1. The function has, at most, n real zeros;

2. The graph has, at most, n – 1 turning points; and

3. Turning points (relative maximum or relative minimum) are points at which the graph changes from increasing to decreasing or vice versa.

​

The zeros of a polynomial function are the values of x which make

f(x) = 0. These values are the roots, or solutions of the polynomial equation when y = 0. All real roots are the x-intercepts of the graph.

​​To find the Zeros of Polynomial Function

​​To find the Zeros of Polynomial Function

​​Behavior of Polynomial functions

The behavior of the graph of a function to the far left and far right is called its end behavior. Although the graph of a polynomial function may have intervals where it increases or decreases, the graph will eventually rise or fall without bound as it moves far to the left or far to the right.

​How can we determine the end behavior of a polynomial function?

​Using the table on the right, observe how the end behavior of the graph changes in relation to the leading coefficient and degree of the polynomial functions

​Word Problems involving Polynomial Functions

​Illustrative Example 5

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Find the length of the edge of a cube, if an edge is increased by 3 dm, another edge has a 6 dm increase and the third one decreases by 2 dm, results to 100% increase of its original volume.

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