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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
3
3.4.1: Extending Polynomial Identities to Include Complex Numbers
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 astatement of the Distributive Property.
Key Concepts, continued
4
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Identity True for…
This is true for all values of a and b,
except for b = –1. The expression
is not defined for 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
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:
6
3.4.1: Extending Polynomial Identities to Include Complex Numbers
r This polynomial has 1 term, so it is called a monomial.
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.
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 one-
variable 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.
8
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 A quadratic equation in
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 number whose square is –1. We define i
so that and i 2 = –1.
• An imaginary number is any number of the form bi,
where b is a real number, , 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.11
3.4.1: Extending Polynomial Identities to Include Complex Numbers
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,
, and b ≠ 0).
• Recall that if x2 = a, then . For example, if
x2 = 25, then x = 5 or x = –5.
12
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,
(Note the use of the negative sign under the radical.)
• For example,
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 For example, if p = 4
and q = 9, then
• But if p and q are both negative, then
For example, if p = –4 and q = –9, then
14
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
• So, to simplify an expression of the form
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:
• 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
(a + bi)(a – bi) = a2 – abi + abi – b2i 2 Distribute.(a + bi)(a – bi) = a2 – b2i 2 Simplify.(a + bi)(a – bi) = a2 – b2(–1) i 2 = –1(a + bi)(a – bi) = a2 + b2 Simplify.
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 for 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 3Write 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 the solutions to x2 + 3.
The quadratic formula is
20
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
21
3.4.1: Extending Polynomial Identities to Include Complex Numbers
x2 + 3 = 0 Set the quadratic polynomial equal to 0.
1x2 + 0x + 3 = 0 Write the polynomial in the form ax2 + bx + c = 0.
Substitute values into the quadratic formula: a = 1, b = 0, and c = 3.
Simplify.
Guided Practice: Example 3, continued
22
3.4.1: Extending Polynomial Identities to Include Complex Numbers
For any positive real number
a,
Factor 12 to show a perfect square factor.
For any real numbers a and
b,
Simplify.
Guided Practice: Example 3, continuedThe solutions of the equation x2 + 3 = 0 are
Therefore, the equation can be written in the
factored form
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.
24
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
25
3.4.1: Extending Polynomial Identities to Include Complex Numbers
x2 + 3 = 0 Set the quadratic polynomial equal to 0.
x2 = –3 Subtract 3 from both sides.
Apply the Square Root
Property: if x2 = a, then
For any positive real number a,
Guided Practice: Example 3, continued
3. Verify the identityby multiplying.
26
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Distribute.
Combine similar terms.
Simplify.
Guided Practice: Example 3, continued
The square root method produces the same result as
the quadratic formula.
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 4Write 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.
The quadratic formula is
30
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
31
3.4.1: Extending Polynomial Identities to Include Complex Numbers
3x2 + 2x + 11 = 0 Set the quadratic polynomial equal to 0.
Substitute values into the quadratic formula: a = 3, b = 2, and c = 11.
Simplify.
Guided Practice: Example 4, continued
32
3.4.1: Extending Polynomial Identities to Include Complex Numbers
For any positive real number
a,
Factor 128 to show its greatest perfect square factor.
For any real numbers a and
b,
Simplify.
Guided Practice: Example 4, continued
The solutions of the equation 3x2 + 2x + 11 = 0 are
33
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Write the real and imaginary parts of the complex number.
Simplify.
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, continuedThe roots of the equation 3x2 + 2x + 11 = 0 are
Therefore, the equation can be written in the
factored form
or in the simpler factored form
35
3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
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