3.6The Real Zeros of Polynomial Functions

Goals:

• Finding zeros of polynomials

• Factoring polynomials completely

Review

divisor

remainderquotient

divisor

dividend

4

19

Review : Synthetic Division

1

)(

x

xf

)()()()( xrxqxdxf

322)( 23 xxxxf

1

)(

x

xf

2. Remainder TheoremRemainder Theorem.For any polynomial f(x) the remainder of is the number

1

322 : ofremainder theDetermine

23

x

xxx

2) Determine the remainder of

1) Determine the remainder of

3. Application of Remainder Theorem

1

22 1835

x

xx

1

2 289

x

xx

4. Recall: Factor Theorem

if and only if

is a factor of

5. Application of Factor Theorem

1242)( 185822 xxxxxf

1) Is x + 1 a factor of ?)(xf

2) Is x - 1 a factor of ?)(xf

6. Factoring of Polynomials

Is a factor of ?

If is a factor of then

20266)( 234 xxxxxf

If yes, then write f(x) in factored form:

)()()( quotientcxxf

)()()( quotientcxxf

)2067)(1( 23 xxxx

summaryIf -3 is a zero of . What does the factor theorem tell us?

1. 2. is a factor of .3. The remainder of is zero4. The point (-3,0) is an x-intercept on the graph.

)(xf

0)3( f

Types of Zeros:

Example of a factored polynomial:

6. Real zeros of a polynomial

Number of Real Zeros Theorem

A polynomial of degree n, has at most n real zeros.

)54)(54)(2)(12)(3()( ixixxxxxf

Rational Zeros Irrational Zeros Complex Imaginary Zeros

6. Real zeros of a polynomial

2 Methods for finding the zeros

1) Graphing calculator

(gives approximation to irrational

zeros)

2) Algebraically

(better for finding exact value of zeros)

6 a) graphing calculator approx.

Graph: p. 184 #81.

x-intercepts: Use ZERO feature y-intercepts: TRACE: x=0

d) Table to determine graph close to zero. Is it above or below?

e) Max/Min

Find zeros (x-intercepts) using graphing calculator.

Rational Roots Theorem Given: a polynomial with integer coefficients.

If has any rational zeros, they will be from the list:

where p = factors of constant term

q = factors of leading coefficient

6b) Identify Rational Zeros

)(xf

q

p

442914)( 23 xxxxfList all possible rational zeros.

8. Test a potential zero

1) graphing calculator (TABLE or Trace)

OR2) Does f(c) = 0 ?

442914)( 23 xxxxf

Finding both rational and irrational zeros.

9. Determine the zeros of a polynomial

284)( :1 Example 45 xxxxf

1) Find zeros on calculator and verify f(c) = 02) How many zeros (x-intercepts) are there?3) Are any zeros repeated?4) Continue synthetic division on previous solution until quotient

is factorable.

2028176)( :2 Example 234 xxxxxf

Example: This function is completely factored

10. Write the complete factorizationWrite as product of:

• linear factorsand

• irreducible quadratic factors

13

2)1()( 22

xxxxf

10. Write the complete factorization

) )(())(()( 21 parteirreduciblcxcxcxxf n

Example 3: 20266)( 234 xxxxxf

1) Find rational zeros

2) Perform synthetic division on each quotient

3) Repeat until reduced to easily factorable quotient .

6. Write the complete factorization

20243132)( :4 Example 234 xxxxxf

Look for repeated zeros (where graph touches at the zero)

6. Write the complete factorization

24446)( :5 Example 2456 xxxxxxf

Reduces to difference of squares that can be factored.

6. Write the complete factorization

5153)( :6 Example 23 xxxxf

If integer zeros are not found on calculator, look for zeros from list of potential zeros. {p/q} and verify.