04/19/2304/19/23 11

Algebra IIAlgebra II

Unit 4Unit 4

Roots and ZerosRoots and Zeros

CCSS: A. APR.3CCSS: A. APR.3

Standards for Mathematical Standards for Mathematical PracticePractice

1. Make sense of problems and persevere in 1. Make sense of problems and persevere in solving them. solving them.

2. Reason abstractly and quantitatively. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the 3. Construct viable arguments and critique the

reasoning of others.  reasoning of others.   4. Model with mathematics. 4. Model with mathematics. 5. Use appropriate tools strategically. 5. Use appropriate tools strategically. 6. Attend to precision. 6. Attend to precision. 7. Look for and make use of structure. 7. Look for and make use of structure. 8. Look for and express regularity in repeated 8. Look for and express regularity in repeated

reasoning.reasoning.

CCSS: A. APR.3CCSS: A. APR.3

Identify Identify zeros of polynomialszeros of polynomials when when suitable factorizations are availablesuitable factorizations are available, , and use the and use the zeroszeros to construct a to construct a rough graphrough graph of the of the function defined function defined by the polynomialby the polynomial..

04/19/2304/19/23 33

ESSENTIAL QUESTION (s):ESSENTIAL QUESTION (s):How do I find the roots of a How do I find the roots of a

polynomial function?polynomial function?

  

How does the Remainder Theorem How does the Remainder Theorem help me to find the roots of a help me to find the roots of a

polynomial function? polynomial function?

  

How do I determine the equation How do I determine the equation and graph of a polynomial and graph of a polynomial function given its roots? function given its roots?

04/19/2304/19/23 44

04/19/2304/19/23 55

Students shall be able toStudents shall be able to Determine and find Determine and find

the number and the number and type of roots for a type of roots for a polynomialpolynomial

Find the zeros of a Find the zeros of a polynomial polynomial equation. equation.

04/19/2304/19/23 66

AgendaAgenda Warm upWarm up Home WorkHome Work LessonLesson PracticePractice

04/19/2304/19/23 77

Homework ReviewHomework Review

04/19/2304/19/23 88

Example 1 Determine Number and Type of Roots

Example 2Find Numbers of Positive and Negative Zeros

Example 3 Use Synthetic Substitution to Find Zeros

Example 4 Use Zeros to Write a Polynomial Function

Concept Summary:Concept Summary:

04/19/2304/19/23 99

on

nn

n axaxaxaxf 1

11

f(c ) = 0c is a zero of the polynomial function f(x)x – c is a factor of polynomial function f(x)c is a root or solution of polynomial function f(x)If c is a real number, then (c,0) is an intercept of the graph f(x)

04/19/2304/19/23 1010

Communicate EffectivelyCommunicate Effectively Theorem:

A proposition that has been or is to be proved on the basis of explicit assumptions.

The Fundamental Theorem of Algebra:

Every polynomial equation with a degree greater than 0 has at least one root in the set of complex numbers. Complex Conjugates Theorem: For every polynomial, if there exists a complex root a+bi then a-bi also exists.

Fundamental Theorem Of AlgebraFundamental Theorem Of Algebra(your text)(your text)

Every Polynomial Equation with a degree higher than zero has at least one root in the set of Complex Numbers.

A Polynomial Equation of the form P(x) = 0 of degree ‘n’ with complex coefficients has exactly ‘n’ Roots in the set of Complex Numbers.

COROLLARY:

Real/Imaginary RootsReal/Imaginary Roots

If a polynomial has ‘n’ complex roots will its graph have ‘n’ x-intercepts?

In this example, the degree n = 3, and if we factor the polynomial, the roots are x = -2, 0, 2. We can also see from the graph that there are 3 x-intercepts.

y x3 4x

Real/ImaginaryReal/Imaginary RootsRoots

Just because a polynomial has ‘n’ complex roots doesn’t mean that they are all Real!

y x3 2x2 x 4In this example, however, the degree is still n = 3, but there is only one Real x-intercept or root at x = -1, the other 2 roots must have imaginary components.

04/19/2304/19/23 1414

Solve State the number and type of roots.

Original equation

Add 10 to each side.

Answer: This equation has exactly one real root, 10.

04/19/2304/19/23 1515

Solve State the number and type of roots.

Original equation

Answer: This equation has two real roots, –8 and 6.

Factor.

Solve each equation.

Zero Product Propertyor

04/19/2304/19/23 1616

Solve State the number and type of roots.

Original equation

Factor out the GCF.

Subtract 6 from each side.

Use the Zero Product Property.

or

04/19/2304/19/23 1717

Square Root Property

Answer: This equation has one real root at 0, and two imaginary roots at

04/19/2304/19/23 1818

Original equation

Solve State the number and type of roots.

Factor differences of squares.

Factor differences of squares.

or or Zero ProductProperty

04/19/2304/19/23 1919

Solve each equation.

Answer: This equation has two real roots, –2 and 2, and two imaginary roots, 2i and –2i.

04/19/2304/19/23 2020

Solve each equation. State the number and type of roots.

a.

b.

c.

Answer: This equation has exactly one root at –3.

Answer: This equation has exactly two roots, –3 and 4.

Answer: This equation has one real root at 0 and two imaginary roots at

04/19/2304/19/23 2121

d.

Answer: This equation has two real roots, –3 and 3, and two imaginary roots, 3i and –3i.

Descartes’ Rule of SignsDescartes’ Rule of Signs

Arrange the terms of the polynomial P(x) in descending degree:

• The number of times the coefficients of the terms of P(x) change sign = the number of Positive Real

Roots (or less by any even number)

• The number of times the coefficients of the terms of P(-x) change sign = the number of Negative

Real Roots (or less by any even number) In the examples that follow, use Descartes’ Rule of Signs to predict the number of + and - Real Roots!

04/19/2304/19/23 2323

State the possible number of positive real zeros, negative real zeros, and imaginary zeros of

Since p(x) has degree 6, it has 6 zeros. However, some of them may be imaginary. Use Descartes Rule of Signs to determine the number and type of real zeros. Count the number of changes in sign for the coefficients of p(x).

yes– to +

yes+ to –

no– to –

no– to –

04/19/2304/19/23 2424

Since there are two sign changes, there are 2 or 0 positive real zeros. Find p(–x) and count the number of sign changes for its coefficients.

no– to –

no– to –

yes– to +

yes+ to –

Since there are two sign changes, there are 2 or 0 negative real zeros. Make a chart of possible combinations.

x 1

04/19/2304/19/23 2525

Answer:

Number of Number of Positive Real Positive Real

ZerosZeros

Number of Number of Negative Real Negative Real

ZerosZeros

Number of Number of Imaginary Imaginary

ZerosZerosTotalTotal

22 22 22 66

00 22 44 66

22 00 44 66

00 00 66 66

04/19/2304/19/23 2626

State the possible number of positive real zeros, negative real zeros, and imaginary zeros of

Answer: The function has either 2 or 0 positive real zeros, 2 or 0 negative real zeros, and 4, 2, or 0 imaginary zeros.

04/19/2304/19/23 2727

Find all of the zeros of

Since f (x) has degree of 3, the function has three zeros. To determine the possible number and type of real zeros, examine the number of sign changes in f (x) and f (–x).

yes yes no

no no yes

04/19/2304/19/23 2828

The function has 2 or 0 positive real zeros and exactly 1 negative real zero. Thus, this function has either 2 positive real zeros and 1 negative real zero or 2 imaginary zeros and 1 negative real zero.

To find the zeros, list some possibilities and eliminate those that are not zeros. Use a shortened form of synthetic substitution to find f (a) for several values of a.

xx 11 ––11 22 44

––33 11 ––44 1414 ––3838

––22 11 ––33 88 ––1212

––11 11 ––22 44 00

Each row in the table shows the coefficients of the depressed polynomial and the remainder.

04/19/2304/19/23 2929

From the table, we can see that one zero occurs at x = –1. Since the depressed polynomial, ,is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

Quadratic Formula

Replace a with 1, b with –2, and c with 4.

04/19/2304/19/23 3030

Simplify.

Simplify.

04/19/2304/19/23 3131

Answer: Thus, this function has one real zero at –1 andtwo imaginary zeros at and The graph of the function verifies that there is only one real zero.

04/19/2304/19/23 3232

Find all of the zeros of

Answer:

04/19/2304/19/23 3333

Short-Response Test Item

Write a polynomial function of least degree with integer coefficients whose zeros include 4 and 4 – i.

Read the Test Item• If 4 – i is a zero, then 4 + i is also a zero, according

to the Complex Conjugate Theorem. So, x – 4, x – (4 – i), and x – (4 + i) are factors of the polynomial function.

04/19/2304/19/23 3434

Solve the Test Item• Write the polynomial function as a product of

its factors.

• Multiply the factors to find the polynomial function.

Write an equation.

Regroup terms.

Rewrite as thedifference of twosquares.

04/19/2304/19/23 3535

Square x – 4 and replace i2 with –1.

Simplify.

Multiply using the Distributive Property.

Combine liketerms.

04/19/2304/19/23 3636

Answer: is a polynomial function of least degree with integral coefficients whose

zeros are 4, 4 – i, and 4 + i.

04/19/2304/19/23 3737

Short-Response Test Item

Write a polynomial function of least degree with integer coefficients whose zeros include 2 and 1 + i.

Answer:

Find Roots/Zeros of a Find Roots/Zeros of a PolynomialPolynomial

We can find the Roots or Zeros of a polynomial by setting the polynomial equal to 0 and factoring.

Some are easier to factor than others!

f (x) x3 4x

x(x2 4)

x(x 2)(x 2)

The roots are: 0, -2, 2

Find Roots/Zeros of a Find Roots/Zeros of a PolynomialPolynomial

If we cannot factor the polynomial, but know one of the roots, we can divide that factor into the polynomial. The resulting polynomial has a lower degree and might be easier to factor or solve with the quadratic formula.

We can solve the resulting polynomial to get the other 2 roots:

f (x) x3 5x2 2x 10

one root is x 5 x 5 x3 5x2 2x 10

x3 5x2

2x 10

2x 10

0

x2 2

(x - 5) is a factor

2, 2x

Find Roots/Zeros of a PolynomialFind Roots/Zeros of a PolynomialIf the known root is imaginary, we can use the Complex Conjugates Theorem.

Ex: Find all the roots of f (x) x3 5x2 7x 51

If one root is 4 - i.

Because of the Complex Conjugate Theorem, we know that another root must be 4 + i.

Can the third root also be imaginary?

Consider… Descartes: # of Pos. Real Roots = 2 or 0

Descartes: # of Neg. Real Roots = 1

Example (con’t)Example (con’t)

Ex: Find all the roots of f (x) x3 5x2 7x 51

If one root is 4 - i.

If one root is 4 - i, then one factor is [x - (4 - i)], and

Another root is 4 + i, & another factor is [x - (4 + i)].

Multiply these factors:

2

2 2

2

2

4 4 4 4 4 4

4 4 16

8 16 ( 1)

8 17

x i x i x x i x i i i

x x xi x xi i

x x

x x

Example (con’t)Example (con’t)

Ex: Find all the roots of f (x) x3 5x2 7x 51

If one root is 4 - i.

x2 8x 17

If the product of the two non-real factors is x2 8x 17

then the third factor (that gives us the neg. real root) is the quotient of P(x) divided by :

x2 8x 17 x3 5x2 7x 51

x3 5x2 7x 51

0

x 3

The third root is x = -3

Finding Roots/Zeros of Finding Roots/Zeros of PolynomialsPolynomials

We use the Fundamental Thm. Of Algebra, Descartes’ Rule of Signs and the Complex Conjugate Thm. to predict the nature of the roots of a polynomial.

We use skills such as factoring, polynomial division and the quadratic formula to find the zeros/roots of polynomials.

Real Life app: PhysiologyReal Life app: Physiology

During a five-second respiratory During a five-second respiratory cycle, the volume of air in liters in cycle, the volume of air in liters in the human lungs can be described by the human lungs can be described by the function:the function:

A(t) = 0.1729t + 0.1522t² - 0.0374t³, A(t) = 0.1729t + 0.1522t² - 0.0374t³, where t is the time in seconds. Find the where t is the time in seconds. Find the volume of air held by the lungs at3 volume of air held by the lungs at3 seconds.seconds.

04/19/2304/19/23 4444

Real Life app: MedicineReal Life app: Medicine Doctors can measure cardiac output in patients at Doctors can measure cardiac output in patients at

high risk for a heart attack by monitoring the high risk for a heart attack by monitoring the concentration of dye injected into a vein near the concentration of dye injected into a vein near the heart. A normal heart dye concentration is heart. A normal heart dye concentration is approximated by:approximated by:

d(x) = - 0.006 x^4 - 0.140x³ - 0.053x² + 1.79x,d(x) = - 0.006 x^4 - 0.140x³ - 0.053x² + 1.79x,

where x is the time in seconds.where x is the time in seconds.

a. Find all real zeros by graphing. Then verify them a. Find all real zeros by graphing. Then verify them by using synthetic division.by using synthetic division.

b. Which root makes sense for an answer to this b. Which root makes sense for an answer to this problem? Why?problem? Why?

04/19/2304/19/23 4545

04/19/2304/19/23 4646

Questions? Questions?

See Agenda for HW.

Work with the practice provided