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 • 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.

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:

Example 1: The Terminology of Polynomial Expressions

The Terminology 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 .

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?

Example 1: The Algebra of Polynomials Add or subtract the polynomials, as indicated. = = = The first step is to identify like terms and group these together. Note that the terms and are similar, as multiplication is commutative. Pull out the and variables and simplify.

Example 2: The Algebra of Polynomials Add or subtract the polynomials, as indicated. = = = Again, the like terms are identified and grouped together after distributing the minus sign over all the terms in the second polynomial.

Example 3: The Algebra of Polynomials Multiply the polynomials, as indicated. = = Use the distributive property first to multiply each term of the first polynomial by each term of the second. None of the resulting terms are similar, so the final answer is a polynomial of 6 terms.

Example 4: The Algebra of Polynomials Multiply the polynomials, as indicated. = = = The four terms that result from the initial multiplication contain two similar terms. We combine these to obtain the final trinomial.

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

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 = + + +

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.

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 Exponents

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.

Example 5: Common Factoring Methods Use the Greatest Common Factor method to factor the following polynomial. = = Applying the distributive property in reverse leads to the factored form of this degree 6 trinomial.

Example 6: Common Factoring Methods Use the Greatest Common Factor method to factor the following polynomials. = = An alternative form of the final answer is . We would have obtained this answer if we had factored out 12a initially. These two answers are equivalent.

Example 7: Common Factoring Methods Use the Greatest Common Factor method to factor the following polynomials. = = = In factoring out the greatest common factor remember that it is being multiplied first by 1 and then by –3. One common error in factoring is to forget factors of 1.

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.

Common Factoring Methods Ex: Use Method 2, factoring by grouping, to factor the following polynomial: = = = Note: the GCF of the four terms is 1, so you cannot use the GCF method. Try rearranging the way they are grouped. Once you group and notice that the two groups each have a GCF: and 1. Don’t forget to factor out 1 from as it is a common mistake to not factor out a 1.

Example 8: Common Factoring Methods Use the Factor by Grouping method to factor the following polynomials. = = = = The GCF of the four terms in the polynomial is 1, so the Greatest Common Factor Method does not directly apply. So we must use Method 2, Factoring by Grouping.

Example 9: Common Factoring Methods Use the Factor by Grouping method to factor the following polynomial. = = = We could also group the first and third terms and the second and fourth terms to obtain the same result. = = = OR

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:

Example 10: Common Factoring Methods Factor the following binomials. = = The first step is to realize that the binomial is a difference of two squares and to identify the two expressions that are being squared. Then follow the pattern to factor the binomial.

Example 11: Common Factoring Methods Factor the following binomials. = = = First recognize the binomial as a sum of two cubes. Then follow the pattern to factor the binomial.

Example 12: Common Factoring Methods Factor the following binomials. = = = In this difference of two cubes the second cube is itself a binomial. But the factoring pattern still applies, leading to the final factored form of the original binomial.

Common Factoring Methods Method 4, Case 1: Leading Coefficient is 1. In this case, p and r will both be 1 so we only need q and s such that . That is, we need two integers whose sum is b, the coefficient of x, and whose product is c, the constant term.

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

Example 13: Common Factoring Methods • 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 –12. Because the product is negative, one integer must be positive and one must be negative. Therefore, the only possibilities are • Additionally, the sum of these two integers must be 1. Therefore, they must be . • Thus,

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.

Example 14: 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.

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:

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

Example 15: Common Factoring Methods Factor the following algebraic expressions: = = = First we factor out . Next, notice that the second factor is a second-degree trinomial and is factorable. In fact, it is a perfect square trinomial.

Example 16: Common Factoring Methods Factor the following algebraic expression: = = Factor out using the properties of exponents to obtain the terms in the second factor. Simplify the second term.

Hawkes Learning Systems: College Algebra