Introduction Identities are commonly used to solve many different types of mathematics problems. In...
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Transcript of Introduction Identities are commonly used to solve many different types of mathematics problems. In...
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.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued
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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
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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.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Key Concepts, continued• Examples of polynomials include:
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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.
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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 A quadratic equation in
this form can have no real solutions, one real solution,
or two real solutions.
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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.
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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.
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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,
(Note the use of the negative sign under the radical.)
• For example,
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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
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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.
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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.
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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.
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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
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice
Example 3Write a polynomial identity that shows how to factor x2 + 3.
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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
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 3, continued
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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.
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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
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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.
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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.
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
✔
Guided Practice: Example 3, continued
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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.
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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
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
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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
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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
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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.
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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
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
Guided Practice: Example 4, continued
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3.4.1: Extending Polynomial Identities to Include Complex Numbers
✔
Guided Practice: Example 4, continued
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3.4.1: Extending Polynomial Identities to Include Complex Numbers