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.
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