Introduction Polynomials are often added, subtracted, and even multiplied. Doing so results in certain sums, differences, or products of polynomials.

Slides:

Advertisements

Similar presentations

Chapter 1 Review In Algebra, we use symbols to stand for various numbers. One type of symbol used is a variable. An expression that contains at least one.

Advertisements

4.5 Complex Numbers Objectives:

Distributive Property

Mathematics made simple © KS Polynomials A polynomial in x is an expression with positive integer powers of x. Degree of Polynomial Terminology 5x is a.

Factoring Polynomials Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Greatest Common Factor The simplest method.

Introduction Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product. We can factor certain expressions.

Introduction Using the Pythagorean Theorem to solve problems provides a familiar example of a relationship between variables that involve radicals (or.

Introduction Polynomial identities can be used to find the product of complex numbers. A complex number is a number of the form a + bi, where a and b are.

Introduction Solving inequalities is similar to solving equations. To find the solution to an inequality, use methods similar to those used in solving.

Introduction Previously, we learned how to solve quadratic-linear systems by graphing and identifying points of intersection. In this lesson, we will focus.

The Distributive Property

Introduction While it may not be efficient to write out the justification for each step when solving equations, it is important to remember that the properties.

Introduction Two equations that are solved together are called systems of equations. The solution to a system of equations is the point or points that.

Introduction Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. We know.

10.1 Adding and Subtracting Polynomials

Introduction Identities are commonly used to solve many different types of mathematics problems. In fact, you have already used them to solve real-world.

Introduction The properties of integer exponents also apply to irrational exponents. In this section, we will see how the following properties can be used.

Introduction As we have already seen, exponential equations are equations that have the variable in the exponent. Some exponential equations are complex.

Algebra 1 Final Exam Review – 5 days (2nd Semester)

Introduction Polynomials, or expressions that contain variables, numbers, or combinations of variables and numbers, can be added and subtracted like real.

The Distributive Property allows you to multiply each number inside a set of parenthesis by a factor outside the parenthesis and find the sum or difference.

Introduction An exponent is a quantity that shows the number of times a given number is being multiplied by itself in an exponential expression. In other.

Introduction To simplify an expression, such as (a + bx)(c + dx), polynomials can be multiplied. Unlike addition and subtraction of polynomial terms, any.

Ch X 2 Matrices, Determinants, and Inverses.

Section 2.1 Solving Equations Using Properties of Equality.

Introduction Completing the square can be a long process, and not all quadratic expressions can be factored. Rather than completing the square or factoring,

5 – 2: Solving Quadratic Equations by Factoring Objective: CA 8: Students solve and graph quadratic equations by factoring, completing the square, or using.

Analyzing Equations and Inequalities Objectives: - evaluate expressions and formulas using order of operations - understand/use properties & classifications.

: Adding and Subtracting Rational Expressions Introduction Expressions come in a variety of types, including rational expressions. A rational expression.

Analyzing Equations and Inequalities Objectives: - evaluate expressions and formulas using order of operations - understand/use properties & classifications.

1.3 – Day 1 Algebraic Expressions. 2 Objectives ► Adding and Subtracting Polynomials ► Multiplying Algebraic Expressions ► Special Product Formulas ►

Introduction The product of two complex numbers is found using the same method for multiplying two binomials. As when multiplying binomials, both terms.

Ch. 6.4 Solving Polynomial Equations. Sum and Difference of Cubes.

1.5 The Distributive Property For any numbers a, b, and c, a(b + c) = ab + ac (b + c)a = ba + ca a(b – c)=ab – ac (b – c)a = ba – ca For example: 3(2 +

Introduction Previously, you learned how to graph logarithmic equations with bases other than 10. It may be necessary to convert other bases to common.

Introduction Polynomials, or expressions that contain variables, numbers, or combinations of variables and numbers, can be added and subtracted like real.

Section 6.2 Solving Linear Equations Math in Our World.

Introduction An exponential function is a function in the form f(x) = a(b x ) + c, where a, b, and c are constants and b is greater than 0 but not equal.

5.3 Notes – Add, Subtract, & Multiply Polynomials.

Introduction Terms of geometric sequences can be added together if needed, such as when calculating the total amount of money you will pay over the life.

The Distributive Property

Introduction A common practice in mathematics is to write answers in simplest form. However, not every expression can be combined or simplified. For example,

Introduction Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product. We can factor certain expressions.

Introduction The properties of integer exponents also apply to irrational exponents. In this section, we will see how the following properties can be used.

Introduction The Pythagorean Theorem is often used to express the relationship between known sides of a right triangle and the triangle’s hypotenuse.

Algebraic Expressions

Introduction The graph of an equation in x and y is the set of all points (x, y) in a coordinate plane that satisfy the equation. Some equations have graphs.

Introduction Polynomials can be added and subtracted like real numbers. Adding and subtracting polynomials is a way to simplify expressions. It can also.

Introduction Expressions can be used to represent quantities when those quantities are a sum of other values. When there are unknown values in the sum,

Using Distributive Property

The Distributive Property

Factoring Polynomials

Introduction While it may not be efficient to write out the justification for each step when solving equations, it is important to remember that the properties.

Introduction As we have already seen, exponential equations are equations that have the variable in the exponent. Some exponential equations are complex.

Introduction Two equations that are solved together are called systems of equations. The solution to a system of equations is the point or points that.

Introduction Let’s say you and your friends draw straws to see who has to do some unpleasant activity, like cleaning out the class pet’s cage. If everyone.

Let’s Review -- An equation is similar to a scale. Both sides of the scale need to be equal in order for the scale to balance. Properties of equality.

Introduction The properties of integer exponents also apply to irrational exponents. In this section, we will see how the following properties can be used.

Rational Exponents, Radicals, and Complex Numbers

Introduction The properties of integer exponents also apply to irrational exponents. In this section, we will see how the following properties can be used.

Introduction Solving inequalities is similar to solving equations. To find the solution to an inequality, use methods similar to those used in solving.

Introduction The graph of an equation in x and y is the set of all points (x, y) in a coordinate plane that satisfy the equation. Some equations have graphs.

Today we will Finish chapter 5

Introduction The product of two complex numbers is found using the same method for multiplying two binomials. As when multiplying binomials, both terms.

Analyzing Equations and Inequalities

Solving Systems of Equations by the Substitution and Addition Methods

Introduction An exponent is a quantity that shows the number of times a given number is being multiplied by itself in an exponential expression. In other.

Introduction So far we have seen a function f of a variable x represented by f(x). We have graphed f(x) and learned that its range is dependent on its.

Factoring Polynomials, Special Cases

Presentation transcript:

Introduction Polynomials are often added, subtracted, and even multiplied. Doing so results in certain sums, differences, or products of polynomials appearing more commonly. Becoming familiar with certain polynomial identities can make the processes of expanding and factoring polynomials easier. 2.2.1: Polynomial Identities

Key Concepts Identities can be used to expand or factor polynomial expressions. A polynomial identity is a true equation that is often generalized so it can apply to more than one example. The tables that follow shows the most common polynomial identities, including the steps of how to work them out. 2.2.1: Polynomial Identities

Key Concepts, continued Common Polynomial Identities Square of Sums Identity Formula Steps With two variables: (a + b)2 = a2 + 2ab + b2 (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 With three variables: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac (a + b + c) 2 = (a + b + c)(a + b + c) = a2 + ab + ac + ab + b2 + bc + ac + bc + c2 = a2 + b2 + c2 + 2ab + 2bc + 2ac (continued) 2.2.1: Polynomial Identities

Key Concepts, continued Square of Differences Identity Formula Steps (a – b)2 = a2 – 2ab + b2 (a – b)2 = (a – b)(a – b) = a2 – ab – ab + b2 = a2 – 2ab + b2 Difference of Two Squares Identity Formula Steps a2 – b2 = (a + b)(a – b) a2 – b2 = a2 + ab – ab + b2 = (a + b)(a – b) (continued) 2.2.1: Polynomial Identities

Key Concepts, continued Sum of Two Cubes Identity Formula Steps a3 + b3 = (a + b)(a2 – ab + b2) a3 + b3 = a3 – a2b + ab2 + a2b – ab2 + b3 = (a + b)(a2 – ab + b2) Difference of Two Cubes Identity Formula Steps a3 – b3 = (a – b)(a2 + ab + b2) a3 – b3 = a3 + a2b + ab2 – a2b – ab2 – b3 = (a – b)(a2 + ab + b2) 2.2.1: Polynomial Identities

Key Concepts, continued Note that the square of sums, (a + b)2, is different from the sum of squares, a2 + b2. Polynomial identities are true for any values of the given variables. When dealing with large numbers, the use of identities often results in simpler calculations. 2.2.1: Polynomial Identities

Common Errors/Misconceptions confusing identities, such as applying the Square of Differences Identity instead of the Difference of Two Squares Identity incorrectly applying an identity to find an equivalent expression misplacing a negative sign in an identity, resulting in an incorrect equivalent expression 2.2.1: Polynomial Identities

Guided Practice Example 1 Use a polynomial identity to expand the expression (x – 14)2. 2.2.1: Polynomial Identities

Guided Practice: Example 1, continued Determine which identity is written in the same form as the given expression. The expression (x – 14)2 is written in the same form as the left side of the Square of Differences Identity: (a – b)2 = a2 – 2ab + b2. Therefore, we can substitute the values from the expression (x – 14)2 into the Square of Differences Identity. 2.2.1: Polynomial Identities

Guided Practice: Example 1, continued Replace a and b in the identity with the terms in the given expression. For the given expression (x – 14)2, let x = a and 14 = b in the Square of Differences Identity. 2.2.1: Polynomial Identities

Guided Practice: Example 1, continued The rewritten identity is (x – 14)2 = x2 – 2(x)(14) + 142. (x – 14)2 Original expression (a – b)2 = a2 – 2ab + b2 Square of Differences Identity [(x) – (14)]2 = (x)2 – 2(x)(14) + (14)2 Substitute x for a and 14 for b in the Square of Differences Identity. 2.2.1: Polynomial Identities

Guided Practice: Example 1, continued Simplify the expression by finding any products and evaluating any exponents. The first term, x2, cannot be simplified further, but the other terms can. 2.2.1: Polynomial Identities

✔ Guided Practice: Example 1, continued When expanded, the expression (x – 14)2 can be written as x2 – 28x + 196. (x – 14)2 = x2 – 2(x)(14) + 142 Equation from step 2 = x2 – 28x + 142 Simplify 2(x)(14). = x2 – 28x + 196 Square 14. ✔ 2.2.1: Polynomial Identities

Guided Practice: Example 1, continued http://www.walch.com/ei/00657 2.2.1: Polynomial Identities

Guided Practice Example 2 Use a polynomial identity to factor the expression x 3 + 125. 2.2.1: Polynomial Identities

Guided Practice: Example 2, continued Determine which identity is written in the same form as the given expression. The first term in the expression, x 3, is a cube. Determine whether the second term in the expression, 125, is also a cube. 125 = 5 • 25 = 5 • 5 • 5 = 53 Since 125 can be rewritten as a cube, the expression x 3 + 125 can be rewritten as x 3 + 53. 2.2.1: Polynomial Identities

Guided Practice: Example 2, continued The identity a3 + b3 = (a + b)(a2 – ab + b2) is in the same form as x 3 + 53. Therefore, the expression x3 + 125, rewritten as x 3 + 53, is in the same form as the left side of the Sum of Two Cubes Identity: a3 + b3 = (a + b)(a2 – ab + b2). 2.2.1: Polynomial Identities

Guided Practice: Example 2, continued Replace a and b in the identity with the terms in the given expression. For the rewritten expression x 3 + 53, let x = a and 5 = b in the Sum of Two Cubes Identity. 2.2.1: Polynomial Identities

Guided Practice: Example 2, continued The rewritten identity is x3 + 53 = (x + 5)[x2 – (x)(5) + 52]. x3 + 53 Rewritten expression a3 + b3 = (a + b)(a2 – ab + b2) Sum of Two Cubes Identity (x)3 + (5)3 = [(x) + (5)][(x)2 – (x)(5) + (5)2] Substitute x for a and 5 for b in the Sum of Two Cubes Identity. 2.2.1: Polynomial Identities

Guided Practice: Example 2, continued Simplify the expression by finding any products and evaluating any exponents. The first factor, x + 5, cannot be simplified further. The second factor, x2 – (x)(5) + 52, has three terms: the first term, x2, cannot be simplified; the other terms, (x)(5) and 52, can be simplified. 2.2.1: Polynomial Identities

✔ Guided Practice: Example 2, continued When factored, the expression x3 + 125, which is equivalent to x3 + 53, can be rewritten as (x + 5)(x2 + 5x + 25). x3 + 53 = (x + 5)[x2 – (x)(5) + 52] Equation from the previous step = (x + 5)(x2 – 5x + 52) Simplify (x)(5). = (x + 5)(x2 + 5x + 25) Square 5. ✔ 2.2.1: Polynomial Identities

Guided Practice: Example 2, continued http://www.walch.com/ei/00658 2.2.1: Polynomial Identities