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.

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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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Key Concepts, continued

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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.


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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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Key Concepts, continued• Examples of polynomials include:

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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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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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• 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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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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• 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


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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• 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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• 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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• 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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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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(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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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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Guided Practice

Example 3Write a polynomial identity that shows how to factor x2 + 3.

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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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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.


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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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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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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,


3. Verify the identityby multiplying.

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Distribute.

Combine similar terms.

Simplify.


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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http://www.walch.com/ei/00365

Guided Practice

Example 4Write a polynomial identity that shows how to factor the polynomial 3x2 + 2x + 11.

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1. Solve for x using the quadratic formula.

The quadratic formula is

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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.


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For any positive real number

a,

Factor 128 to show its greatest perfect square factor.

For any real numbers a and

b,

Simplify.


The solutions of the equation 3x2 + 2x + 11 = 0 are

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Write the real and imaginary parts of the complex number.

Simplify.


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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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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introduction identities are commonly used to solve many different types of mathematics problems. in...

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