Polynomial Functions

Intro to Polynomial Functions

Polynomial Function

• graph of a polynomial function is always a smooth curve – no breaks, holes, or corners

• Leading Coefficient indicates end behavior

• terms combined by add/subtract/multiply/divide (but can't divide by a variable... exponents must be positive whole numbers)

• Can have constant, variables, and exponents

• Can have many terms, but not an infinite number of terms

• Domain is ALL REAL NUMBERS

Multiple Select

Which are the following are considered polynomials?

• $x^3-2x^2+3x-6$

• $\sqrt{x+6}$

• $x^2+x^{-3}$

• $xy^3+x^2-6$

• $\frac{3x-5}{x+7}$

Degree

• The degree of a polynomial in one variable is determined by the highest exponent of the variable in the expression.

• Leading coefficient is the numerical coefficient of the leading term.

• Polynomial with one variable ... degree is equal to the largest exponent

• Polynomial with more than one variable ... degree is equal to the largest sum of the exponents with the largest sum

• Constant term is the term that do not contains variable.

Degree, Leading Term, Leading Coefficient, and Constant Term

• The degree of a polynomial is the highest power of the said polynomial. If the powers are not arranged, look for the highest exponent.

• Leading term is the term having the highest exponent.

• Leading Coefficient is the constant (number) in the leading term.

•  Constant Term term is the term that has no variables, only a number.

Multiple Choice

What is the leading coefficient?

 $5x^3-4x^2+6x-7$  

• 4

• 5

• 6

• 7

Multiple Choice

What is the constant term?

 $4x^3-2x^2-5x+8$  

•  $4x^3$  

•  $-2x^2$  

•  $-5x$  

• 8

Multiple Choice

Determine the degree of the polynomial.

 $5x^4-3x^3+6x^2-x+7$  

• 2

• 3
  

• 4

• 5

Multiple Choice

Determine the degree of the polynomial.

 $8x^4y^2-3x^3y+6x^2y-x+1$  

• 8

• 6
  

• 4

• 2

Standard Form

• written starting with highest degree and continuing with terms in descending order according to degree

• if any power of x is missing, then it has a coefficient of zero

Multiple Choice

When a polynomial is written in standard form, the coefficient of the first term is called the _____________.

• constant

• leading coefficient

• first number

• degree

Multiple Select

Select all the functions that are written in standard form.

•  $x-3x^2+6$  

•  $x^6-3x^3+2x^2-5$  

•  $x^5-x^2+x^4+3x-2$  

•  $x^5+2x^4-3x+6$  

Classifying Polynomials

• Classified by degree

• Classified by number of terms

Multiple Select

Classify the polynomial by degree and number of terms.  Select the 2 correct options.

 $5x^4+4x^2-3x+7$  

• quartic 

• quintic

• trinomial

• polynomial

Multiple Select

Classify the polynomial by degree and number of terms.  Select the 2 correct options.

 $6x^3+5x^2+x$  

• cubic

• quintic

• trinomial

• polynomial

Multiple Choice

What is the end behavior of the graph?

 $4x^3-3x+2$  

• Up - up

• Up - down

• Down- down

• Down - up

Multiple Choice

What is the end behavior of the graph?

 $-4x^2-3x+2$  

• Up - up

• Up - down

• Down- down

• Down - up

Multiple Choice

What is the degree?

• odd

• even

Multiple Choice

How many turning points does this graph have?

• 0

• 1

• 2

• 3

Multiple Choice

What is the leading coefficient?

• positive

• negative

Multiple Choice

This function has 3 turning points. What is the minimum degree of this function?

• 1

• 2

• 3

• 4

Zeros of a Function are also known as:

• Roots

• Solutions

• X-Intercepts

This function has 3 zeros.

x = - 1

x = 1

x = 4

Multiple Choice

Identify all of the zeros of the given function.

• $x=2$

• $x=-3$

• $x=5$

• all the above

Multiple Choice

Which is NOT a zero of this function?

• $x=-3$

• $x=-6$

• $x=3$

Finding Zeros of a Function from its Equation or Factored Form (DESMOS)

• Type either the function, equation, or factored form into Desmos

• Move cursor to points where the graph crosses the x axis

Multiple Select

Check all that are zeros of the function.

• $-170$

• $-3$

• 1

• 4

• 40

Finding Zeros from Factored Form

• Make sure that f (x) = 0

• Once factored form is set equal to zero, use Zero Product Property

• Solve each factor for x to find each real zero (x-intercept)

Multiple Select

Find the zeros.

 $\left(2x-3\right)\left(x+5\right)$  

•  $x=\frac{2}{3}$  

•  $x=\frac{3}{2}$  

•  $x=5$  

•  $x=-5$  

Multiple Select

List the zeros. Check all that apply.

 $x\left(x-7\right)\left(x+3\right)=0$  

• 0

•  $-3$  

• 3

• 7

•  $-7$  

Writing the Factored Form given the Zeros

• Start with the factored form P(x) = a (x - r₁)(x - r₂)(x - r_n)

• Find each factor by inserting each zero into the factor form (x - r) where r is the real zero (x-intercept).

• The answer can be left with a generic "a" in the factored form. If another point from the function is known, the "a" can be calculated.

Multiplicity of Zeros

• Multiplicity tells us how the graph behaves at the zeros

• Odd Multiplicity: Graph will cross the x-axis through the x-intercept

• Even Multiplicity: Graph will touch the x-axis at the x-intercept

Multiple Choice

What is the multiplicity of $\left(x-1\right)$  in the polynomial  $P\left(x\right)=\ \left(x-1\right)^3\left(x+2\right)$  

• 1

• 2

• 3

Multiple Choice

How does the graph behave around the zero $\left(x-1\right)$  in the polynomial  $P\left(x\right)=\ \left(x-1\right)^3\left(x+2\right)$  ?

• touches the x-axis

• crosses the x-axis

• doesn't cross or touch the x-axis

Multiple Choice

What is the multiplicity of x-intercept $x=2$  ?

• even multiplicity

• odd multiplicity

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