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Chapter 7: Polynomials - Addition, Subtraction, and Multiplication

This chapter introduces polynomials and covers addition, subtraction, and multiplication of polynomials. Topics include terms, like terms, types of polynomials, factoring, and algebraic modeling. Key concepts are reviewed to prepare for the chapter.

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Chapter 7: Polynomials - Addition, Subtraction, and Multiplication

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  1. Chapter 7: Polynomials This chapter starts on page 320, with a list of key words and concepts.

  2. Chapter 7: Get Ready! • Here are the concepts that need to be reviewed before starting Chapter 7: • Represent expressions using algebra tiles. • The zero principle • Polynomials • Factors

  3. 7.1 Add and Subtract Polynomials! • A term is an expression formed by the product of numbers and variables. • 3x2 and 4x are examples of terms.

  4. What is a variable? • A variable is a letter that is used to represent a value that can change or vary. • For example, in 4x – 1, the variable is x.

  5. The parts of a term • There are 2 parts of a term: • The numerical coefficient • The literal coefficient Think: numerical=number literal=letter

  6. The numerical coefficient • The numeric factor of a term is called the numerical coefficient. • For example, the numerical coefficient of 4x is 4 and for -3y2 it is -3

  7. The literal coefficient • The non-numeric factor (i.e. the letter) of a term is called the literal coefficient. • For example, the literal coefficient of 4x is x and for -3y2 it is y2. (You have to include the exponent)

  8. A polynomial • A polynomial is an algebraic expression consisting of one or more terms separated by addition (+) or subtraction (-) symbols.

  9. Types of polynomials • There are 4 different types of polynomials: • Monomials • Binomials • Trinomials • Polynomials

  10. The definition of each polynomial • A monomial has one term. • A binomial has two terms. • A trinomial has three terms. • A polynomial is the general expression used when there is more than one term.

  11. Like terms • Like terms are terms that have the same literal coefficient. • For example, 3x and 4x are like terms because they have the same literal coefficient, x. • And -3y2 and 5y2 are like terms

  12. An algebraic model • An algebraic model can represent a pattern, a relationship or a numeral sequence. • An algebraic model is always written in the form of an algebraic expression, algebraic equation or algebraic formula.

  13. 7.3: Multiply a monomial by a polynomial • Here is the distributive property, a rule that allows you to simplify expressions involving the multiplication of a monomial by a polynomial. • 3(x + 2) = 3(x) + 3(2) = 3x + 6

  14. The expansion of expressions • When you apply the distributive property, you are expanding an expression.

  15. 7.4: Multiply two binomials • In order to multiply 2 binomials, we can use: • Area models using Alge-Tiles. • Multiply using distributive property

  16. The area of a rectangle Area of a rectangle = length of rectangle x width of rectangle

  17. Method #1 (Area models) • When building rectangular tile models, use these directions: • Begin at the bottom left corner with x2 tiles first. • Construct a rectangle in the top right corner with unit tiles. • Fill the top left and bottom right spaces with x-tiles.

  18. Method #2 Distributive Property • In order to use this method properly, use these directions: • Multiply the 2 first terms together in each bracket • Multiply the first term in the first bracket with the second term in the second bracket • Multiply the second term in the first bracket with the first term in the second bracket together • Multiply the 2 last terms together • Add all the products together in order to obtain the simplified expression. (Combine like terms!)

  19. The result of multiplying 2 binomials • When you multiply 2 binomials together, you will get a trinomial *** • For example: • (x + 2)(x + 3) = x2 + 5x + 6

  20. 7.5: Polynomial Division • To divide a polynomial by a monomial, it is like applying the distributive property in reverse. • For example, (6x + 9) ÷ 3 = (6x/3) + (9/3) = 2x + 3 • *** A number divided by itself equals 1. (4÷4=1 and x÷x=1)

  21. 7.2: Common Factors • There are 3 ways to factor a polynomial: • The sharing model • The area model • The greatest common factor method (GCF)

  22. Factoring a polynomial • In order to factor a polynomial completely, find the polynomial’s greatest common factor. • You can find these common factors in the numerical coefficients, in the literal coefficients or in the both of them.

  23. Which method should you use? • The sharing model works best when the common factor is a number. • The area model works best when the common factor is a letter.

  24. An example of factoring • 3x + 12 = 3(x + 4) • 3x + 12 = 3(x + 4) are equivalent expressions.

  25. The expanded form • 3x + 12 is in the expanded form and contains two terms.

  26. The factored form • 3(x + 4) is in the factored form. • The factored form has 2 types of factors: 3 is the common numeric factor and (x + 4) is the polynomial factor.

  27. 7.6: Applying algebraic modeling • Here is how you can solve an algebraic word problem: • Read the problem at least 3 times. • Identify the known and unknown quantities. • Make a plan that will solve for the unknown quantities. • Solve your problem with the plan that you came up with in #3. • Write your final answer in a complete sentence.

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