2.3 Real Zeros of Polynomial Functions
Pre-calculus.
Name: ________________________________
Date: ________________________ Block: ___
1. Long Division of Polynomials.
We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using various methods. Long
division is another method of factoring that is especially useful for polynomials of higher degree.
Consider 6x 3 −19 x 2 +16x − 4 . Do we any algebraic method for factoring this polynomial?
Suppose you are given that one of the ____________ is (x − 2) . This means that f (2) = ____ , and that x = 2 is a _________
of the polynomial function. This also means there is an ________________________ at (2,0) .
This means that there exists a second-degree polynomial q(x) such that f (x) = (x − 2)⋅ q(x) .
To find q(x) you can use long division.
Divide 6x 3 −19 x 2 +16x − 4 by x − 2 . Use the result to factor the polynomial completely.
6x 3 −19x 2 +16x − 4
=
x −2
)
First re-write the fraction in long division form as x − 2 6x 3 −19x 2 +16x − 4
The divisor is x − 2 , the dividend is 6x −19 x +16x − 4 .
3
2
quotient
divisor)dividend
Notice that the terms are written so that the exponents are in descending order. Verify that all placeholders are written.
1. Divide the first term of the dividend by the first term of the divisor. The result is the first term of the quotient.
2. Multiply the first term of the quotient by the first term of the divisor. The result is the first term of the new dividend.
3. Multiply the first term of the quotient by the second term of the divisor. The result is the second term of the new
dividend.
4. Subtract the original dividend from the new dividend. Bring down the next term from the original dividend.
5. Repeat #1-4 with the new dividend until you arrive at the remainder.
6x 2 − 7x + 2
x − 2 6x 3 −19x 2 +16x − 4
)
6x 3 −12x 2
− 7x 2 +16x
− 7x 2 +14 x
2x − 4
2x − 4
0
The quotient is the result of dividing the dividend by the divisor. For this problem, the quotient is 6x 2 − 7x + 2 .
Now you have the original factor (x − 2) and another factor of degree two 6x 2 − 7x + 2 .
Therefore we know that
6x 3 −19x 2 +16x − 4
= 6x 2 − 7x + 2
x −2
Can you factor 6x 2 − 7x + 2 further?
6x 3 −19x 2 +16x − 4
= (2x −1)(3x − 2) Therefore the complete factorization is (x − 2)(2x −1)(3x − 2) . Graph and verify.
x −2
In the previous example we used long division and the result had a remainder of zero. Often, long division will produce a
nonzero remainder.
Example 2: Divide x 2 + 3x + 5 by x +1. Label the dividend, divisor, quotient, and remainder.
x 2 + 3x + 5
3
= x +2+
The result can be written as
. Label the dividend, divisor, quotient, and remainder.
x +1
x +1
This implies that x 2 + 3x + 5 = (x +1)(x + 2) + 3 which illustrates the Division Algorithm.
The division algorithm is commonly written in two ways:
f (x) = d(x)q(x) + r(x)
and
f (x)
r(x)
= q(x) +
d(x)
d(x)
This means that if f (x) and d(x) are polynomials such that d(x) ≠ 0 , and the degree of d(x) is less than or equal to the
degree of f (x) , there exist unique polynomials q(x) and r(x) where r(x) = 0 or the degree of r(x) is less than the degree
of d(x) . If the remainder r(x) is zero, d(x) divides evenly into f (x)
Example 3: Divide x 3 −1 by x −1. Be careful to account for missing terms with zero coefficients or empty spaces.
x 3 −1
= x 2 + x +1,
Answer:
x −1
x ≠ 1 You can always check your answer by multiplying the resulting quotient by the divisor
which will equal the original dividend.
(x 2 + x +1)(x −1) =
Example 4: Divide 2x 4 + 4 x 3 − 5x 2 + 3x − 2 by x 2 + 2x − 3 .
Answer:
2x 4 + 4 x 3 − 5x 2 + 3x − 2
x +1
= 2x 2 +1+ 2
2
x + 2x − 3
x + 2x − 3
Individual Practice: try these on your own paper.
1. x + 3 x 2 + 10 x + 21
2. 3x 2 − 2 x 6 x 4 + 5 x 3 + 3x − 5
3. x + 5 x 2 + 8 x + 15
4. x − 2 x 2 + 3x − 10
5. x + 2 x 3 + 5 x 2 + 7 x + 2
6. x − 3 x 4 − 81
7. 3x 2 − x − 3 6 x 3 + 13 x 2 − 11x − 15
8. x 2 + x − 2 x 4 + 2 x 3 − 4 x 2 − 5 x − 6
10. 2 x 3 + 1 2 x 5 − 8 x 4 + 2 x 3 + x 2
9. 3x 2 + 1 18 x 4 + 9 x 3 + 3x 2
11. What does it mean if the remainder is zero?
12. If a 6th degree polynomial is divided by a 3rd
degree polynomial, what is the degree of the
quotient?
Recap.
We already know if x = a is a zero, then ________________ is the corresponding factor of that zero.
f ( x)
= q ( x) , where q(x) is the quotient.
x−a
If (x − a) is a factor and we divide f (x) by (x − a) , then the remainder, r(x) = _______.
If the degree of f is n , what is the degree of q(x) ? _____________
We also know that we can divide
•
•
2. Synthetic Division.
Synthetic division can be used when dividing by a linear factor in the form ________________________.
There is a pattern you must know when dividing a polynomial by a linear factor using synthetic division.
The vertical pattern is addition; the diagonal pattern is multiplication by k.
Example 1: Use synthetic division to divide x 4 −10x 2 − 2x + 4 by x + 3 . Write your answer in the form
f (x)
r(x)
= q(x) +
d(x)
d(x)
−3 1 0 −10 −2 4
Individual practice: Use synthetic division to divide the following:
1. Divide x 4 − 2 x 3 + 4 x 2 − 2 x + 5 by x − 3.
2. ( x 3 + 512) ÷ ( x + 8)
3.
5x 3 + 6x + 8
x +2
3. Remainder & Factor Theorems.
The Remainder Theorem: if a polynomial f (x) is divided by x − k , the remainder is r = f (k) . The Remainder Theorem tells
you that synthetic division can be used to evaluate a polynomial function when x = k by dividing f (x) by x − k . The
remainder will be f (k) .
Example 1: Use the remainder theorem to evaluate the function f (x) = 3x 3 + 8x 2 + 5x − 7 at x = −2 .
Because the remainder r = _____, you can conclude that f (____) = ____ . This also means that the ordered pair
(____,____) is a point on the graph of f .
Example 2: Find the remainder if f (x) = x 3 − 4 x 2 + 2x − 5 is divided by:
a. (x − 3)
b. (x + 2)
Individual practice: Use synthetic division to find each function value. Use a graphing calculator to verify your results.
1. f (x) = 4 x 3 −13x +10
2. p(x) = x 6 − 4 x 4 + 3x 2 + 2
a. f (1)
b. f (−2)
 1
 2
c. f  
d. f (8) a. p(2)
b. p(−4)
c. p(3)
d. p(−1)
The Factor Theorem: a polynomial f (x) has a factor (x − k) if and only if f (k) = 0 . In other words, to determine whether a
polynomial has (x − k) as a factor, evaluate the polynomial at x = k . If the result is 0, (x − k) is a factor.
Example 2: Show that (x − 2) and (x + 3) are factors of f (x) = 2x 4 + 7x 3 − 4 x 2 − 27x −18 . Find the remaining factors of
f (x) . Use synthetic division.
Answer: f (x) = (x − 2)(x + 3)(2x + 3)(x +1)
Example 3: Is x + 4 a factor of f (x) = 4 x 6 − 64 x 4 + x 2 −15 ?
Individual practice: Verify the given factors of f . Find the remaining factors of f . Use your results to write the complete
factorization of f . List all the real zeros of f . Confirm your results with a graphing calculator.
1. f (x) = 2x 3 + x 2 − 5x + 2 , given the factors (x + 2), (x −1)
2. f (x) = 3x 3 + 2x 2 −19x + 6 ,
(x + 3), (x − 2)
Using the remainder in synthetic division.
In summary, the remainder r , obtained in the synthetic division of f (x) by x − k , provides the following information.
1. The remainder r gives the value of f at x = k . That is, r = f (k) .
2. If r = 0 , (x − k) is a factor of f (x) .
3. If r = 0 , (k,0) is an x-intercept of the graph of f .
4. The Rational Zero Test relates the possible rational zeros of a polynomial (having integer coefficients) to the leading
coefficient and to the constant term of the polynomial. In other words, if a polynomial has integer coefficients, every rational
zero of f has the form
p
where p and q have no common factors other than 1, p is a factor of the constant term, and q is a
q
factor of the leading coefficient.
Possible rational zeros = ±
From the list of possible rational zeros we can use trial-and-error to determine which (if any) are actual zeros of the
polynomial.
Example 1: Find the rational zeros of f (x) = x 3 + x +1
If there are no rational zeros, what must be true?
Example 2: Find the rational zeros of f (x) = 2x 3 + 3x 2 − 8x + 3 using the rational zero test. Use synthetic division to verify
whether the possible zero is an actual zero of the polynomial.
Example 3: Find all the real zeros of f (x) = 10x 3 −15x 2 −16x +12 using the rational zero test and synthetic division to
verify whether the possible zeros are actual zeros of the polynomial.
Individual practice: Use the rational zero test to list all possible rational zeros of f . Use a graphing calculator to verify that
the zeros of f are contained in the list.
1. f (x) = x 3 − 4 x 2 − 4 x +16
2. f (x) = 4 x 5 − 8x 4 − 5x 3 +10x 2 + x − 2
3. f (x) = 3x 5 + x 4 + 4 x 3 − 2x 2 + 8x − 5
4. f ( x) = 2 x 3 + 11x 2 − 7 x − 6
How do we find the zeros?
• Rational Zero Test: gives us a list of possible rational zeros. To determine whether the possible zeros are actual zeros you
may use synthetic or long division.
• The Remainder Theorem and Factor Theorem give us a quick way to determine if an input is a zero of the function.
• Synthetic division is a short cut for dividing linear factors.
• The degree of a polynomial gives the maximum number of zeros.
Steps for finding the zeros of a polynomial:
1. Use the degree to determine the maximum number of zeros.
2. If the polynomial has integer coefficients, use the Rational Zero Test to identify potential rational zeros.
3. Test each zero using the Remainder Theorem (or Factor Theorem).
4. Once you find a zero, use synthetic division to break down the polynomial. Keep testing for zeros and use the depressed
equation (quotient) to keep breaking down the polynomial until all zeros are found.
* Factor out the GCF of a polynomial before you begin.
** If you are able to find all but two of the zeros, you can always use factoring or the quadratic
formula to find the remaining two zeros.
*** Note: Not all zeros are rational. Some may be irrational and some may be complex (imaginary).
1. Let f ( x) = 2 x 3 + 3x 2 − 8 x + 3 . Find the real zeros and write the polynomial in factored form (as a product of linear
factors.)
3
2
2. Let g ( x) = 6 x − 4 x + 3x − 2 . Find the real zeros and write the polynomial in factored form.
3. Let g(x) = x 5 − 5x 4 +12x 3 − 24 x 2 + 32x −16 . Find the real zeros and write the polynomial in factored form (as a
product of linear factors.)