FACTORING POLYNOMIALS

In this course we will study a number of factoring techniques used to identify the factors of certain polynomials. They are: 1. Greatest Common Monomial Factor 2. Grouping 3. Difference of Two Squares 4. Trinomial type 1 (x2 + bx + c) 5. Trinomial type 2 (ax2 + bx + c) An important facet in learning how to factor is identifying when to apply each technique. The biggest clue to this is in the number of terms that the original polynomial contains. We will focus on this issue later when we have learned several techniques.

Greatest Common Monomial Factor Let’s break this down to identify exactly what this is. First of all a factor is basically a divisor because all of those numbers can divide evenly into it. Factors of 12  1, 2, 3, 4, 6, 12 Factors of 12  1, 2, 3, 4, 6, 12 Factors of 18  1, 2, 3, 6 ,9, 18 Factors of 12  1, 2, 3, 4, 6, 12 Factors of 18  1, 2, 3, 6 ,9, 18 Factors of 18  1, 2, 3, 6 ,9, 18 Common Factors for 12 and 18  1, 2, 3, 6 Common Factors for 12 and 18  1, 2, 3, 6 Greatest Common Factor for 12 and 18  6 A monomial factor is a one term factor. This expression is sometimes used when finding factors of polynomials.

It is very useful to know the prime factors of a given number It is very useful to know the prime factors of a given number. This refers to the prime numbers that are multiplied to produce the given number. Examples: 15 = 3 × 5 39 = 3 × 13 12 = 2 × 2 × 3 42 = 2 × 3 × 7 16 = 2 × 2 × 2 × 2 84 = 2 × 2 × 3 × 7 75 = 3 × 5 × 5 112 = 2 × 2 × 2 × 2 × 7 Notice that some of the given numbers are expressed with 2 factors, others with 3 factors and yet others with more than 3. All of the factors are prime numbers. Prime numbers are the most basic factors you can get.

Therefore: 225 = 3 × 5 × 3 × 5 OR 225 = 3 × 3 × 5 × 5 An easy way to determine the prime factors of a number is by constructing a factor tree where the given number is the trunk of the tree and the prime factors are the ends of the branches. 72 72 8 9 6 12 2 4 3 2 3 2 6 2 × 2 × × 3 × 3 2 × 3 2 2 3 Therefore: 72 = 2 × 2 × 2 × 3 × 3 Notice that no matter which two factors that you begin with, you end up with the same prime factors. 225 15 3 5 × 3 5 Therefore: 225 = 3 × 5 × 3 × 5 OR 225 = 3 × 3 × 5 × 5

Polynomial expressions with 2 or more terms Just as numbers have factors so do polynomial expressions. The factors of monomial expressions are pretty obvious, however the factors of polynomial expressions having 2 or more terms are not so obvious. For this reason it is necessary to learn techniques that illuminate the factors of these kinds of polynomial expressions. Monomial expressions 2x = 2  x = 2  2  3  3  3  p  p  q 108p2q 24a2bc3 = 2  2  2  3  a  a  b  c  c  c Polynomial expressions with 2 or more terms 108p3q2 + 144p3q + 81p2q2 + 108p2q m2 - 17mn + 72n2 9a2 – 15ab – 6ac – 6ab2 + 10b3 + 4b2c – 5a + 5ab

Given the polynomial expression: 24x4y3z7 + 16x2y5z4 - 44xz6 Each of these 3 terms have many factors. They have factors that are common to all 3 terms – many, in fact. Ex: 1, 2, 4, x, z, xz, 4z2, xz3, 4xz4 However, they have only 1 greatest common factor which is a monomial term. (4xz4) We can divide each of the terms of the polynomial by this GCF.

By dividing each of the terms of the polynomial by this GCF, rewrite the equivalent value of the polynomial as 2 separate factors. One factor being the GCF monomial and the other being the leftover terms after division. Leftover terms GCF ( 6x3y3z3 + 4xy5 - 11z2 ) 4xz4 If we multiply 4xz4 by 6x3y3z3 + 4xy5 - 11z2 then we will arrive back at the polynomial that we started with.  24x4y3z7 + 16x2y5z4 - 44xz6 This is a way to verify that we have correctly factored the polynomial.

Factor 3x2 + 6xy When determining the GCF of a polynomial, we can take it in steps. 1. Determine the GCF of the numerical coefficients. GCF of 3 and 6 is 3 2. Determine the GCF of each variable. We can do this by choosing the lowest power of a given variable from each term. GCF of x2 and x is x because x is the variable that has lowest exponent of those 2 (exponent is 1). With no y-variable in the first term and y in the second term there can’t be a common factor containing variable y.

Factor: 3x2 + 6xy GCF = 3x Factors: 3x(x + 2y) Factor: 10x4y3 + 4x3y2 - 2x2y2 GCF of 10, 4 and 2 is 2 GCF of x4, x3 and x2 is x2 because x2 is the variable that has lowest exponent of those 3 (exponent is 2). GCF of y3, y2 and y2 is y2 because y2 is the variable that has lowest exponent of those 3 (exponent is 2). GCF = 2x2y2

Factor: 10x4y3 + 4x3y - 2x2y2 GCF = 2x2y2 Factors: 2x2y2(5x2y2 + 2x - 1) Factor: -3m3n – 7m3r + 8m3rt GCF of 3, 7 and 8 is 1 GCF of m3, m3 and m3 is m3 because each term has the variable m3. Only one term has the variable n so it is not common to all terms Not all terms have the variable r so it is not common to all terms Only one term has the variable t so it is not common to all terms

Factor: -3m3n – 7m3r + 8m3rt GCF = 1m3 Because the leading term (-3m3n) is negative, we will make the common factor negative. GCF = -1m3 Factors: -m3(3n + 7r – 8rt) Factor: -2ab3 – 4b3c – 12b3d GCF of 2, 4 and 12 is 2 Only one term has the variable a so it is not common to all terms GCF of b3, b3 and b3 is b3 because each term has the variable b3. Not all terms have the variable c so it is not common to all terms Only one term has the variable d so it is not common to all terms

Factor: -2ab3 – 4b3c – 12b3d GCF = -2b3 Because the leading term (-2ab3 ) is negative, we will make the common factor negative. GCF = -2b3 Factors: -2b3 (a + 2c + 6d)

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