Download 2.3.3

Introduction
Synthetic division, along with your knowledge of end
behavior and turning points, can be used to identify the
x-intercepts of a polynomial function. This information
allows for more accurate sketches of functions.
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2.3.3: Finding Zeros
Key Concepts
• Recall that roots are the x-intercepts of a function. In other
words, these are the x-values for which a function equals 0.
These zeros are another way of referring to the roots of a
function.
• When a polynomial equation with a degree greater than 0 is
solved, it may have one or more real solutions, or it may
have no real solutions (in which case it would have complex
solutions).
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2.3.3: Finding Zeros
Key Concepts
• Recall that both real and imaginary numbers belong to the
set of complex numbers; therefore, all polynomial functions
with a degree greater than 0 will have at least one root in the
set of complex numbers. This is referred to as the
Fundamental Theorem of Algebra.
Fundamental Theorem of Algebra
If p(x) is a polynomial function of degree n ≥ 1 with
complex coefficients, then the related equation
p(x) = 0 has at least one complex solution (root).
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2.3.3: Finding Zeros
Key Concepts, continued
• A repeated root is a root that occurs more than once
in a polynomial function.
• Recall that the solutions to a quadratic equation that
contains imaginary numbers come in pairs. These are
called complex conjugates, the complex number that
when multiplied by another complex number produces
a value that is wholly real; the complex conjugate of
a + bi is a – bi.
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2.3.3: Finding Zeros
Key Concepts, continued
•
If an imaginary number is a zero of a function, its
conjugate is also a zero of that function. This is true
for all polynomial functions, and is known as the
Complex Conjugate Theorem.
Complex Conjugate Theorem
Let p(x) be a polynomial with real coefficients. If
a + bi is a root of the equation p(x) = 0, where a
and b are real and b ≠ 0, then a – bi is also a root
of the equation.
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2.3.3: Finding Zeros
Key Concepts, continued
•
For a polynomial function p(x), the factor of a
polynomial is any polynomial that divides evenly into
that function. Recall that when a polynomial is
divided by one of its factors, there is a remainder of
0 and the result is a depressed polynomial. This is
an illustration of the Factor Theorem.
Factor Theorem
The binomial x – a is a factor of the polynomial
p(x) if and only if p(a) = 0, where a is a real
number.
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2.3.3: Finding Zeros
Key Concepts, continued
• The Factor Theorem can help to find all the factors of
a polynomial.
• To do this, first show that the binomial in question is a
factor of the polynomial. If the remainder is 0, then the
binomial is a factor.
• Then, determine if the resulting depressed polynomial
can be factored.
• The identified factors indicate where the function
crosses the x-axis.
• The zeros of a function are related to the factors of
the polynomial. The graph of a polynomial function
shows the zeros of the function, which are the
x-intercepts of the graph.
2.3.3: Finding Zeros
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Key Concepts, continued
• It is often helpful to know which integer values of a to
try when determining p(a) = 0.
• Use the Integral Zero Theorem to determine the
zeros of a polynomial function.
• Identify the factors of the constant term of a
polynomial function and use substitution to determine
if each number results in a zero.
• Use synthetic division to determine the remaining
factors.
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2.3.3: Finding Zeros
Key Concepts, continued
Integral Zero Theorem
If the coefficients of a polynomial function are
integers such that an = 1 and a0 ≠ 0, then any
rational zeros of the function must be factors of a0.
• For example, consider the equation p(x) = x2 + 10x + 25.
an = 1 and a0 = 25. A possible zero of this function is –5
because –5 is a factor of 25.
• The zeros of any polynomial function correspond to the
x-intercepts of the graph and to the roots of the
corresponding equation.
2.3.3: Finding Zeros
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Key Concepts, continued
• If a polynomial has a factor x – a that is repeated n
times, then the root is called a repeated root and x = a
is a zero of multiplicity. Multiplicity refers to the
number of times a zero of a polynomial function
occurs.
• If the multiplicity is odd, then the graph intersects the
x-axis at the point (x, 0). If the multiplicity is even, then
the graph just touches the axis at the point (x, 0).
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2.3.3: Finding Zeros
Key Concepts, continued
• If p(x) is a polynomial with real coefficients whose terms
are arranged in descending powers of the variable, then
the number of positive zeros of y = p(x) is the same as the
number of sign changes of the coefficients of the terms, or
is less than this by an even number. Recall that when
solving a quadratic function, we used the quadratic
formula, which generated pairs of roots. Often, these roots
were complex and the x-intercepts could not be graphed
on the coordinate plane. For this reason, we must count
down from our maximum number of zeros by twos to
determine the number of real zeros. For example, for a
polynomial with 3 sign changes, the number of positive
zeros may be 3 or 1, since 3 – 2 = 1.
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2.3.3: Finding Zeros
Key Concepts, continued
• Also, the number of negative zeros of y = p(x) is the
same as the number of sign changes of the
coefficients of the terms of p(–x), or is less than this
number by an even number.
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2.3.3: Finding Zeros
Common Errors/Misconceptions
• not finding all the factors of a polynomial function
• making sign errors when performing synthetic division
• misusing the terms “roots” and “factors”
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2.3.3: Finding Zeros
Guided Practice
Example 1
Given the equation x3 + 4x2 – 3x – 18 = 0, state the
number and type of roots of the equation if one root
is –3.
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2.3.3: Finding Zeros
Guided Practice: Example 1, continued
1. Use synthetic division to find the
depressed polynomial.
The related polynomial is x3 + 4x2 – 3x – 18, with
coefficients 1, 4, –3, and –18.
One root of the equation is –3; therefore, a factor of
the related polynomial is [x – (–3)], which simplifies
to x + 3.
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2.3.3: Finding Zeros
Guided Practice: Example 1, continued
Divide the related polynomial by x + 3 to find the
depressed polynomial. Let the value of a be –3.
-3
1
1
4
-3
- 18
-3
-3
18
1
-6
0
The depressed polynomial is x2 + x – 6.
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2.3.3: Finding Zeros
Guided Practice: Example 1, continued
2. Factor the depressed polynomial to find
the remaining factors.
Use previously learned strategies to factor the
polynomial.
x2 + x – 6
Depressed polynomial
(x – 2)(x + 3)
Factor the polynomial.
The remaining factors are (x – 2) and (x + 3).
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2.3.3: Finding Zeros
Guided Practice: Example 1, continued
3. State the number of roots and type of
roots of the equation.
The factors of the related polynomial are (x + 3),
(x – 2), and (x + 3). Recall that we divided the
depressed polynomial by (x + 3) before finding the
remaining factors, so we must include (x + 3) as a
factor. Therefore, the roots of the equation are –3, –3,
and 2. Since –3 appears twice, it is a repeated root.
Because of the repeated root, this equation
has only 2 real roots: –3 and 2.
✔
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2.3.3: Finding Zeros
Guided Practice: Example 1, continued
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2.3.3: Finding Zeros
Guided Practice
Example 3
Write the simplest polynomial function with integral
coefficients that has the zeros 5 and 3 – i.
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2.3.3: Finding Zeros
Guided Practice: Example 3, continued
1. Determine additional zeros of the
function.
Because 3 – i is a zero, then according to the
Complex Conjugate Theorem, the conjugate 3 + i is
also a zero.
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2.3.3: Finding Zeros
Guided Practice: Example 3, continued
2. Use the zeros to write the polynomial as a
product of the factors.
The zeros are 5, 3 – i, and 3 + i. These can be written
as the factors x – 5, x – (3 – i), and x – (3 + i).
The polynomial function written in factored form with
zeros 5 and 3 – i is f(x) = (x – 5)[x – (3 – i)][x – (3 + i)].
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2.3.3: Finding Zeros
Guided Practice: Example 3, continued
3. Multiply the factors to determine the
polynomial function.
Polynomial function
f(x) = (x – 5)[x – (3 – i )][x – (3 + i )] written in factored
form
f(x) = (x – 5)[(x – 3) – i ][(x – 3) + i ] Regroup terms.
f(x) = (x – 5)[(x – 3)2 – i 2]
Rewrite as the
difference of two
squares.
f(x) = (x – 5)(x2 – 6x + 9 – i 2)
Simplify.
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2.3.3: Finding Zeros
Guided Practice: Example 3, continued
f(x) = (x –
5)[x2
– 6x + 9 – (–1)]
Replace i 2 with –1,
since i 2 = –1.
f(x) = (x – 5)(x2 – 6x + 9 + 1)
Simplify.
f(x) = x3 – 11x2 + 40x – 50
Distribute.
The polynomial function of least degree with integral
coefficients whose zeros are 5 and 3 – i is
f(x) = x3 – 11x2 + 40x – 50.
✔
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2.3.3: Finding Zeros
Guided Practice: Example 3, continued
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2.3.3: Finding Zeros