Section 2-4

Real Zeros of Polynomial Functions

Section 2-4

• long division and the division algorithm

• the remainder and factor theorems

• reviewing the fundamental connection for polynomial functions

• synthetic division

• rational zeros theorem

• upper and lower bounds

Long Division

• long division for polynomials is just like long division for numbers

• it involves a dividend divided by a divisor to obtain a quotient and a remainder

• the dividend is the numerator of a fraction and the divisor is the denominator

Division Algorithm

• if f(x) is the dividend, d(x) is the divisor, q(x) the quotient, and r(x) the remainder, then the division algorithm can be stated two ways

( ) ( ) ( ) ( )

or

( ) ( )( )

( ) ( )

f x d x q x r x

f x r xq x

d x d x

Remainder Theorem• if a polynomial f (x) is divided by x – k, then

the remainder is f (k)• in other words, the remainder of the division

problem would be the same value as plugging in k into the f (x)

• we can find the remainders without having to do long division

• later, we will find f (k) values without having to plug k into the function using a shortcut for long division

Factor Theorem• the useful aspect of the remainder theorem is

what happens when the remainder is 0

• since the remainder is 0, f (k) = 0 which means that k is a zero of the polynomial

• it also means that x – k is a factor of the polynomial

• if we could find out what values yield remainders of 0 then we can find factors of polynomials of higher degree

Fundamental Connection

• k is a solution (or root) of the equation f (x) = 0

• k is a zero of the function f (x)

• k is an x-intercept of the graph of f (x)

• x – k is a factor of f (x)

For a real number k and a polynomial function f (x), the following statements are equivalent

Synthetic Division• finding zeros and factors of polynomials

would be simple if we had some easy way to find out which values would produce a remainder of 0 (long division takes too long)

• synthetic division is just that shortcut

• it allows us to quickly divide a function f (x) by a divisor x – k to see if it yields a remainder of 0

Synthetic Division

• it follows the same steps as long division without having to write out the variables and other notation

• it is really fast and easy

• if a zero is found, the resulting quotient is also a factor, and it is called the depressed equation because it will be one degree less than the original function

Synthetic Division3 2Divide 2 3 5 12 by 3x x x x

Synthetic Division3 2Divide 2 3 5 12 by 3x x x x

3 2 - 3 - 5 - 12

Synthetic Division3 2Divide 2 3 5 12 by 3x x x x

3 2 - 3 - 5 - 12

2

6

3

9

4

12

0

Synthetic Division3 2Divide 2 3 5 12 by 3x x x x

3 2 - 3 - 5 - 12

2

6

3

9

4

12

0

The remainder is 0 so x – 3 is a factor and the quotient, 2x2 + 3x + 4, is also a factor

Rational Zeros Theorems• if you want to find zeros, you need to

have an idea about which values to test in S.D. (synthetic division)

• the rational zeros theorem provides a list of possible rational zeros to test in S.D.

• they will be a value where,

p must be a factor of the constant q must be a factor of the leading term

pq

Finding Possible Rational Zeros

3 2

Find all the possible rational zeros of

( ) 3 4 5 2f x x x x

2 and 3

factors of p: 1, 2

factors of q: 1, 3

1 2possible rational zeros: 1, 2, ,

3 3

p q

Upper and Lower Bounds

• a number k is an upper bound if there are no zeros greater than k; if k is plugged into S.D., the bottom line will have no sign changes

• a number k is a lower bound if there are no zeros less than k; if k is plugged into S.D., the bottom line will have alternating signs (0 can be considered + or -)

Upper and Lower Bounds

• if you are looking for zeros and you come across a lower bound, do not try any numbers less than that number

• if you are trying to find zeros and you come across an upper bound, do not try any numbers greater than that number