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Hawkes Learning Systems: College Algebra

Hawkes Learning Systems: College Algebra. Section 1.5: Polynomials and Factoring. Objectives:. The terminology of polynomial expressions The algebra of polynomials Common factoring methods. The Terminology of Polynomial Expressions.

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Hawkes Learning Systems: College Algebra

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  1. Hawkes Learning Systems: College Algebra Section 1.5: Polynomials and Factoring

  2. Objectives: • The terminology of polynomial expressions • The algebra of polynomials • Common factoring methods

  3. The Terminology of Polynomial Expressions • Coefficient: A number multiplied by a variable in any of the terms of a polynomial. • Degree of the term: The sum of the exponents of the variables in that term. • Constant term: Any non-zero number that is not multiplied by a variable. Note: Constant terms have a degree of zero. • Degree of a polynomial: The largest degree of all the individual terms.

  4. The Terminology of Polynomial Expressions • Monomials: Polynomials consisting of a single term Ex: • Binomials: Polynomials consisting of two terms Ex: • Trinomials: Polynomials consisting of three terms Ex:

  5. Example: The Terminology of Polynomial Expressions

  6. Polynomials of a Single Variable Polynomials of a Single Variable A polynomial in the variable of a degree n can be written in the form where are real numbers, and n is a positive integer. This form is called descending order because the powers descend from left to right. The leading coefficient of this polynomial is .

  7. Example: Descending Order Write the polynomial in descending order. State the degree of the polynomial and the leading coefficient. 4 – 2x5 + x2

  8. The Algebra of Polynomials Like or similar terms: The terms among all the polynomials being added that have the same variables raised to the same powers. Ex: What are the like terms in the polynomial below? Notice that and both include the variable x raised to the third power. These are like or similar terms. Can you find any others?

  9. Example: Algebra of Polynomials • Subtract the Polynomials. (x4 + 2x3 – x + 5) – (x3 – x – x4) b) Multiply the Polynomials. (x2 – 2y)(x2 + y)

  10. The Algebra of Polynomials When a binomial is multiplied by a binomial, the acronym FOIL is commonly used as a reminder of the four necessary products. Consider the product: The solution to the product above would be First + Outer + Inner + Last First Outer Inner Last

  11. The Algebra of Polynomials Consider the product: The product of the First terms is The product of the Outer terms is The product of the Inner Terms is The product of the Last terms is So, First + Outer + Inner + Last = + = 3a2b2 + 2a3b – a4

  12. Common Factoring Methods: Terminology • Factoring:Reversing the process of multiplication in order to find two or more expressions whose product is the original expression. • Factorable: A polynomial with integer coefficients is factorable if it can be written as a product of two or more polynomials, all of which also have integer coefficients. • Irreducible (over the integers) or prime:A polynomial that is not factorable. • Completely Factor: To write a polynomial as a product of prime polynomials.

  13. Common Factoring Methods • Method 1: Greatest common factor. • Method 2: Factoring by grouping. • Method 3: Factoring special binomials. • Method 4: Factoring trinomials. • Case 1: Leading coefficient is 1. • Case 2: Leading coefficient is not 1. • Method 5: Factoring Expressions Containing Fractional or Negative Exponents

  14. Common Factoring Methods: Method 1 Find the greatest common factor of GCF: So, Method 1: Greatest Common Factor The Greatest Common Factor (GCF) among all the terms is simply the product of all the factors common to each.The Greatest Common Factor method is a matter of applying the distributive property to “un-distribute” the greatest common factor. What do the three terms in the polynomial have in common? . The product of these terms is the GCF of the polynomial.

  15. Common Factoring Methods: Method 2 Method 2: Factoring by Grouping. Factoring by Grouping: A trial and error processapplied when the first factoring method is not directly applicable. If the terms of the polynomial are grouped in a suitable way, the GCF method may apply to each group, and a common factor might subsequently be found among the groups.

  16. Example: Factoring by Grouping. Factor by Grouping. ax – 2bx – 2ay + 4by

  17. Common Factoring Methods • Method 3: Factoring Special Binomials Three types of binomials can always be factored following certainpatterns. In the following, A and B represent algebraic expressions. • Difference of Two Squares: • Difference of Two Cubes: • Sum of Two Cubes:

  18. Example: Factoring Special Binomials Factor the Special Binomials. • 49a2 – 144b2 • x3 – 27y3 • m6 + 125n9

  19. Common Factoring Methods Method 4, Case 1: Leading Coefficient is 1. In this case, we only need q and s such that = (x + q)(x + s) That is, we need two integers whose sum is b, the coefficient of x, and whose product is c, the constant term. Consider (x + 7)(x + 2) =

  20. Common Factoring Methods Use Method 4, Case 1 to factor the following polynomial. Ex: • Factor: • Begin by writing • We need to find two integers to replace the question marks. The two integers we seek must have a product of 2. Because the product is positive, both integers must be either positive or negative. Therefore, the only possibilities are • Additionally, the sum of these two integers must be 3. Therefore, they must be • Thus, or

  21. Example: Factoring Trinomials Factor the Trinomial. • x2 + 2x – 15 • 6x2 + 5x – 6

  22. Common Factoring Methods Method 4, Case 2: Leading Coefficient is not 1. Factoring Trinomials by Grouping For the trinomial : Step 1: Multiply a and c. Step 2: Factor ac into two integers whose sum is b. If no such factors exist, the trinomial is irreducible over the integers. Step 3: Rewrite b in the trinomial with the two integers found in step 2. The resulting polynomial of four terms may now be factored by grouping.

  23. Example: Common Factoring Methods Factor the following trinomial by grouping: 1. 2. 3. 4. 5. Multiply a and c. Factor ac into two integers whose sum is b. Rewrite b in the trinomial with the two integers found in step 2 and distribute. Group.

  24. Common Factoring Methods • Perfect Square Trinomials: trinomial expressions whose factored form is the square of a binomial expression. • There are two forms of Perfect Square Trinomials:

  25. Example: Perfect Square Trinomials Factor the Trinomial. y2 + 14y + 49

  26. Common Factoring Methods: Method 5 Method 5: Factoring Expressions Containing FractionalorNegative Exponents To factor an algebraic expression that has fractional or negative exponents, identify the least exponent among the various terms and factor the variable raised to that least exponent from each of the terms.

  27. Example: Factoring with Negative Exponents Factor the algebraic expression. 7a-1 – 2a-3b

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