Download Key Concepts, continued

Introduction
Identities are commonly used to solve many different
types of mathematics problems. In fact, you have
already used them to solve real-world problems. In this
lesson, you will extend your understanding of polynomial
identities to include complex numbers and imaginary
numbers.
1
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts
• An identity is an equation that is true regardless of
what values are chosen for the variables.
• Some identities are often used and are well known;
others are less well known. The tables on the next two
slides show some examples of identities.
2
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
Identity
True for…
x+2=2+x
This is true for all values of x.
This identity illustrates the
Commutative Property of
Addition.
a(b + c) = ab + ac
This is true for all values of a,
b, and c. This identity is a
statement of the Distributive
Property.
3
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
Identity
ab + a
b +1
=a
True for…
This is true for all values of a and b,
except for b = –1. The expression
ab + a
= a is not defined for b = –1
b +1
because if b = –1, the denominator is
equal to 0. To see that the equation is true
provided that b ≠ –1, note that
ab + a a(b + 1) a (b + 1)
=
=
= a.
b +1
(b + 1)
(b + 1)
3.4.1: Extending Polynomial Identities to Include Complex Numbers
4
Key Concepts, continued
• A monomial is a number, a variable, or a product of a
number and one or more variables with whole number
exponents.
• If a monomial has one or more variables, then the
number multiplied by the variable(s) is called a
coefficient.
• A polynomial is a monomial or a sum of monomials.
The monomials are the terms, numbers, variables, or
the product of a number and variable(s) of the
polynomial.
5
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• Examples of polynomials include:
r
2
3
This polynomial has 1 term, so
it is called a monomial.
x -x
5
This polynomial has 2 terms,
so it is called a binomial.
3x2 – 5x + 2
This polynomial has 3 terms,
so it is called a trinomial.
–4x3y + x2y2 – 4xy3
This polynomial also has 3
terms, so it is also a trinomial.
6
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• In this lesson, all polynomials will have one variable.
The degree of a one-variable polynomial is the
greatest exponent attached to the variable in the
polynomial. For example:
• The degree of –5x + 3 is 1. (Note that
–5x + 3 = –5x1 + 3.)
• The degree of 4x2 + 8x + 6 is 2.
• The degree of x3 + 4x2 is 3.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• A quadratic polynomial in one variable is a onevariable polynomial of degree 2, and can be written in
the form ax2 + bx + c, where a ≠ 0. For example, the
polynomial 4x2 + 8x + 6 is a quadratic polynomial.
• A quadratic equation is an equation that can be written
in the form ax2 + bx + c = 0, where x is the variable, a,
b, and c are constants, and a ≠ 0.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• The quadratic formula states that the solutions of a
quadratic equation of the form ax2 + bx + c = 0 are
given by x =
-b ± b 2 - 4ac
. A quadratic equation in
2a
this form can have no real solutions, one real solution,
or two real solutions.
9
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• In this lesson, all polynomial coefficients are real
numbers, but the variables sometimes represent
complex numbers.
• The imaginary unit i represents the non-real value
. i is the
i =number
-1 whose square is –1. We define i so
that
andi =
i 2 =-1
–1.
• An imaginary number is any number of the form bi,
where b is a real number, i = -1 , and b ≠ 0.
10
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• A complex number is a number with a real
component and an imaginary component. Complex
numbers can be written in the form a + bi, where a
and b are real numbers, and i is the imaginary unit.
For example, 5 + 3i is a complex number. 5 is the real
component and 3i is the imaginary component.
• Recall that all rational and irrational numbers are real
numbers. Real numbers do not contain an imaginary
component.
3.4.1: Extending Polynomial Identities to Include Complex Numbers
11
Key Concepts, continued
• The set of complex numbers is formed by two distinct
subsets that have no common members: the set of
real numbers and the set of imaginary numbers
(numbers of the form bi, where b is a real number,
i = -1, and b ≠ 0).
• Recall that if x2 = a, then x = ± a . For example, if
x2 = 25, then x = 5 or x = –5.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• The square root of a negative number is defined
such that for any positive real number a, -a = i a.
(Note the use of the negative sign under the radical.)
• For example,
-9 = i 9 = i · 3 = 3i.
13
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• Using p and q as variables, if both p and q are
positive, then p · q = pq. For example, if p = 4
and q = 9, then p · q = 4 · 9 = 2 · 3 = 6, and
pq = 4 · 9 = 36 = 6.
• But if p and q are both negative, then p · q ¹ pq.
For example, if p = –4 and q = –9, then p · q =
-4 · -9 = i 4 · i 9 = 2i · 3i = 6i 2 = 6 · (-1) = -6, but
pq =
( -4) ( -9) =
36 = 6.
14
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• So, to simplify an expression of the form p · q
when p and q are both negative, write each factor as a
product using the imaginary unit i before multiplying.
15
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• Two numbers of the form a + bi and a – bi are called
complex conjugates.
• The product of two complex conjugates is always a
real number, as shown:
(a + bi)(a – bi) = a2 – abi + abi – b2i 2
(a + bi)(a – bi) = a2 – b2i 2
(a + bi)(a – bi) = a2 – b2(–1)
(a + bi)(a – bi) = a2 + b2
Distribute.
Simplify.
i 2 = –1
Simplify.
• Note that a2 + b2 is the sum of two squares and it is a
real number because a and b are real numbers.
16
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• The equation (a + bi)(a – bi) = a2 + b2 is an identity
that shows how to factor the sum of two squares.
17
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Common Errors/Misconceptions
• substituting pq for p · q when p and q are both
negative
• neglecting to include factors of i when factoring the sum
of two squares
18
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice
Example 3
Write a polynomial identity that shows how to factor x2 + 3.
19
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
1. Solve for x using the quadratic formula.
x2 + 3 is not a sum of two squares, nor is there a
common monomial.
Use the quadratic formula to find the2 solutions to
-b ± b - 4ac
x2 + 3.
x=
.
2a
The quadratic formula is
20
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
Set the quadratic polynomial
equal to 0.
2
x +3=0
2
1x + 0x + 3 = 0
x=
x=
-0 ± 0 2 - 4 (1) ( 3 )
2 (1)
± -12
2
Write the polynomial in the
2
form ax + bx + c = 0.
Substitute values into the
quadratic formula: a = 1,
b = 0, and c = 3.
Simplify.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
x=
x=
x=
x=
±i 12
2
±i · 4 · 3
2
±i · 4 · 3
2
±2i 3
2
For any positive real number
a, -a = i a.
Factor 12 to show a perfect
square factor.
For any real numbers a and
b, ab = a · b.
Simplify.
x = ±i 3
3.4.1: Extending Polynomial Identities to Include Complex Numbers
22
Guided Practice: Example 3, continued
The solutions of the equation x2 + 3 = 0 are i 3 and
-i 3.
Therefore, the equation can be written in the
(
)(
)
factored form x + i 3 x - i 3 = 0.
(
)(
)
x 2 + 3 = x + i 3 x - i 3 is an identity that shows
how to factor the polynomial x2 + 3.
23
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
2. Check your answer using square roots.
Another method for solving the equation x2 + 3 = 0 is
by using a property involving square roots.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
2
x +3=0
2
x = –3
Set the quadratic polynomial
equal to 0.
Subtract 3 from both sides.
Apply the Square Root
x = ± -3
x = ±i 3
2
Property: if x = a, then
x = ± a.
For any positive real number a,
-a = i a.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
3. Verify the identity
by multiplying.
(x + i
(x + i
(x + i
(x + i
)(
3 )( x - i
3 )( x - i
3 )( x - i
)
3) = x - i ( 3)
3 ) = x - ( -1) ( 3 )
3) = x + 3
3 x - i 3 = x - xi 3 + xi 3 - i
2
2
2
2
( 3)
2
2
2
Distribute.
Combine
similar terms.
Simplify.
2
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
The square root method produces the same result as
the quadratic formula.
(
)(
)
x 2 + 3 = x + i 3 x - i 3 is an identity that shows
how to factor the polynomial x2 + 3.
✔
27
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
28
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice
Example 4
Write a polynomial identity that shows how to factor the
polynomial 3x2 + 2x + 11.
29
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
1. Solve for x using the quadratic formula.
x=
The quadratic formula is
-b ± b 2 - 4ac
2a
.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
2
3x + 2x + 11 = 0
Set the quadratic polynomial
equal to 0.
-2 ± 22 - 4 ( 3 ) (11) Substitute values into the
x=
quadratic formula: a = 3,
2 ( 3)
b = 2, and c = 11.
x=
x=
-2 ± 4 - 132
6
Simplify.
-2 ± -128
6
3.4.1: Extending Polynomial Identities to Include Complex Numbers
31
Guided Practice: Example 4, continued
x=
x=
x=
x=
-2 ± i 128
6
-2 ± i · 64 · 2
6
-2 ± i · 64 · 2
6
-2 ± 8i 2
For any positive real number
a, -a = i a.
Factor 128 to show its
greatest perfect square
factor.
For any real numbers a and
b, ab = a · b.
Simplify.
6
3.4.1: Extending Polynomial Identities to Include Complex Numbers
32
Guided Practice: Example 4, continued
-2 8i 2
x=
±
6
6
Write the real and imaginary
parts of the complex
number.
1 4 2
x=- ±
i
3
3
Simplify.
The solutions of the equation 3x2 + 2x + 11 = 0 are
1 4 2
1 4 2
- +
i and - i.
3
3
3
3
33
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
2. Use the solutions from step 1 to write the
equation in factored form.
If (x – r1)(x – r2) = 0, then by the Zero Product
Property, x – r1 = 0 or x – r2 = 0, and x = r1 or x = r2.
That is, r1 and r2 are the roots (solutions) of the
equation.
Conversely, if r1 and r2 are the roots of a quadratic
equation, then that equation can be written in the
factored form (x – r1)(x – r2) = 0.
34
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
The roots of the equation 3x2 + 2x + 11 = 0 are
1 4 2
1 4 2
- +
i and - i.
3
3
3
3
Therefore, the equation can be written in the
é
æ 1 4 2 öùé
æ 1 4 2 öù
factored form ê x - ç - +
i÷ ú êx - ç - i ÷ ú = 0,
3 øúê
3 øú
êë
è 3
è 3
ûë
û
or in the simpler factored form
æ
1 4 2 öæ
1 4 2 ö
i÷ ç x + +
i ÷ = 0.
çx+ 3
3 øè
3
3 ø
35
è
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
æ
1 4 2 öæ
1 4 2 ö
3x + 2x + 11= ç x + i÷ ç x + +
i ÷ is an identity
3
3 øè
3
3 ø
è
2
that shows how to factor the polynomial 3x 2 + 2x + 11.
✔
36
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
37
3.4.1: Extending Polynomial Identities to Include Complex Numbers