Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Introduction A variety of concepts, systems, and processes can be described using polynomial expressions. Some of these include familiar geometrical ideas such as area and perimeter. Other real-world applications involve more complex expressions, like the paths followed by baseballs, or satellites in orbit. Factoring, or writing the expressions as the product of their factors, can make the expressions easier to work with and apply to different scenarios. One of the most common forms of factoring is by the greatest common factor, or GCF. 1 3A.2.2: Factoring Expressions by the Greatest Common Factor Key Concepts • Recall that a monomial is an expression with one term, consisting of a number, a variable, or the product of a number and variable(s), whereas a polynomial is an expression with two or more terms, consisting of a number, a variable, or the product of a number and variable(s). • The factors of a given number or monomial are numbers or expressions which multiply together to equal that given number, or monomial. For example, 6 and 3 are factors of 18, and x and x2 are factors of x3. Two possible factors of the monomial 15x4y are 5xy and 3x3. 3A.2.2: Factoring Expressions by the Greatest Common Factor 2 Key Concepts, continued • A common factor is a factor that two or more numbers share, or the same number or variable that appears in all terms of a polynomial. You can find common factors by listing the factors for each term and then comparing the results. For example, consider the polynomial 36x4 + 27x3y. One common factor of 36 and 27 is 3, since 3 • 12 = 36 and 3 • 9 = 27. As for the variables, x is a common factor found in both terms. Notice that the variable y does not appear in both terms of the polynomial, so it is not a common factor. Thus, a common factor of 36x4 + 27x3y is 3x. 3 3A.2.2: Factoring Expressions by the Greatest Common Factor Key Concepts, continued • The greatest common factor (GCF) of a polynomial is the common factor with the largest constant number and power of each variable. In other words, the GCF is the largest factor that two or more terms share. • The GCF for the polynomial 36x4 + 27x3y can be found by first breaking down each term by listing all of its prime factors. Prime factors are factors that are prime numbers, which are whole numbers that can only be divided evenly by 1 and the number itself. 4 3A.2.2: Factoring Expressions by the Greatest Common Factor Key Concepts, continued • For instance, the prime factors of the terms of the polynomial 36x4 + 27x3y are as follows: 36x4 = 2 • 2 • 3 • 3 • x • x • x • x 27x3y = 3 • 3 • 3 • x • x • x • y • To determine the GCF of this polynomial, determine which factors each term has in common. Each term has two prime factors of 3 in common, and each term also has three factors of x: 36x 4 = 2 • 2 • 3 • 3 • x • x • x • x 27x 3 y = 3 • 3 • 3 • x • x • x • y 5 3A.2.2: Factoring Expressions by the Greatest Common Factor Key Concepts, continued • Therefore, the GCF of the polynomial 36x4 + 27x3y is 3 • 3 • x • x • x = 9x3. • Factoring is the process of rewriting a given expression as the product of its factors. Once the GCF is identified, then the remaining factors are determined by multiplying the GCF by the constants and/or variables necessary to obtain the original term: 9x3 • 4x = 36x4 9x3 • 3y = 27x3y 6 3A.2.2: Factoring Expressions by the Greatest Common Factor Key Concepts, continued • The completed factoring for the polynomial expression 36x4 + 27x3y is 9x3(4x + 3y). • To verify that the factors have been correctly identified, apply the Distributive Property to the factored expression. The result should be the original polynomial. • Distributing 9x3 over (4x + 3y) yields 9x3 • 4x + 9x3 • 3y = 36x4 + 27x3y, which verifies that the identified factors are correct. • An expression that cannot be factored is said to be prime. 7 3A.2.2: Factoring Expressions by the Greatest Common Factor Common Errors/Misconceptions • misidentifying the greatest common factor • incorrectly applying a negative sign when factoring or applying the Distributive Property • incorrectly applying the rules of exponents; e.g., x2 • x3 ≠ x6 • including factors that are common to some terms but not to all terms 8 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice Example 1 Factor the polynomial 12x3 + 30x2 + 42x by finding the greatest common factor (GCF). Verify your results using the Distributive Property. 9 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 1, continued 1. Determine the common factors of the terms’ coefficients and variables. First factor the coefficients (12, 30, and 42) into their prime factors. The prime factors are as follows: 12 = 2 • 2 • 3 30 = 2 • 3 • 5 42 = 2 • 3 • 7 10 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 1, continued Each coefficient’s factors include 2 and 3, so 2 • 3 = 6 is common to all three terms. Next, determine the greatest power of x common to all three terms. The first term has x3, the second term has x2, and the third term has x (or x1). Expand these terms: x3 = x • x • x x2 = x • x x1 = x Only x (or x to the first power) is common to all three terms. Combining this with the greatest numerical factor of 6, the resulting GCF is 6x. 11 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 1, continued 2. Write expressions for the remaining non-common factors. Write out the prime factors of each term: 12x3 = 2 • 2 • 3 • x • x • x 30x2 = 2 • 3 • 5 • x • x 42x = 2 • 3 • 7 • x 12 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 1, continued Eliminate the common factors of each term and determine which non-common factors are left: 12x3 = 2 • 2 • 3 • x • x • x ; the remaining factors are 2 • x • x, or 2x2 30x 2 = 2 • 3 • 5 • x • x ; the remaining factors are 5 • x, or 5x 42x = 2 • 3 • 7 • x ; the remaining factor is 7 The expressions for the remaining non-common factors are 2x2, 5x, and 7. 13 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 1, continued 3. Factor the original polynomial by writing it as the product of the GCF and a polynomial whose terms are the expressions found in the previous step. The GCF is 6x. The remaining terms are 2x2, 5x, and 7, which can be summed to produce the polynomial 2x2 + 5x + 7. The product of the GCF and this polynomial is 6x(2x2 + 5x + 7). 14 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 1, continued 4. Verify the result using the Distributive Property. To verify the result, apply the Distributive Property by multiplying 6x by each term of the polynomial. 6x(2x2 + 5x + 7) = 6x • 2x2 + 6x • 5x + 6x • 7 = 12x3 + 30x2 + 42x The result of applying the Distributive Property is the original expression, 12x3 + 30x2 – 42x. Thus, the polynomial is factored correctly and 12x3 + 30x2 + 42x = 6x(2x2 + 5x + 7). ✔ 15 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 1, continued 16 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice Example 2 Factor the polynomial 35xy4 – 14x4y2z + 56x3y3z3 by finding the GCF. Verify your results using the Distributive Property. 17 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 2, continued 1. Determine the common factors of the coefficients for each term. First factor the coefficients (35, –14, and 56) into their prime factors: 35 = 5 • 7 –14 = –1 • 2 • 7 56 = 2 • 2 • 2 • 7 Note that a negative number, like –14, will always have a negative factor that can be represented by –1. The only numerical factor that is common to all three terms is 7. 3A.2.2: Factoring Expressions by the Greatest Common Factor 18 Guided Practice: Example 2, continued 2. Determine the greatest power of each variable common to all terms, and multiply this by the common factor from step 1 to complete the GCF. The first term has the variables xy4, the second has variables x4y2z, and the third has variables x3y3z3. Expand these terms: xy4 = x • y • y • y • y x4y2z = x • x • x • x • y • y • z x3y3z3 = x • x • x • y • y • y • z • z • z 19 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 2, continued One factor of x and two factors of y, or y2, are common to all three terms. Since z is not a factor of the first term, it is not a common factor. Multiplying these common variables by the result of step 1 yields a GCF of 7xy2. 20 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 2, continued 3. Write expressions for the remaining non-common factors. Write out the prime factors of each term: 35xy4 = 5 • 7 • x • y • y • y • y –14x4y2z = –1 • 2 • 7 • x • x • x • x • y • y • z 56x3y3z3 = 2 • 2 • 2 • 7 • x • x • x • y • y • y • z • z • z Eliminate the common factors of each term and determine which non-common factors are left: 35xy 4 = 5 • 7 • x • y • y • y • y The remaining factors are 5 • y • y, or 5y2. 3A.2.2: Factoring Expressions by the Greatest Common Factor 21 Guided Practice: Example 2, continued –14x 4 y 2z = –1 • 2 • 7 • x • x • x • x • y • y • z The remaining factors are –1 • 2 • x • x • x • z, which simplifies to –2x3z. 56x3 y 3z3 = 2 • 2 • 2 • 7 • x • x • x • y • y • y • z • z • z The remaining factors are 2 • 2 • 2 • x • x • y • z • z • z, which simplifies to 8x2yz3. The expressions for the remaining non-common factors are 5y2, –2x3z, and 8x2yz3. 22 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 2, continued 4. Factor the original polynomial by writing it as the product of the GCF and a polynomial whose terms are the expressions found in the previous step. The GCF is 7xy2. The remaining terms are 5y2, –2x3z, and 8x2yz3, which can be summed to produce the polynomial 5y2 – 2x3z + 8x2yz3. The product of the GCF and this polynomial is 7xy2(5y2 – 2x3z + 8x2yz3). 23 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 2, continued 5. Verify the result using the Distributive Property. 7xy2(5y2 – 2x3z + 8x2yz3) = 7xy2 • 5y2 – 7xy2 • 2x3z + 7xy2 • 8x2yz3) = 35xy4 – 14x4y2z + 56x3y3z3 The result of applying the Distributive Property is the original expression, 35xy4 – 14x4y2z + 56x3y3z3. Thus, the polynomial is factored correctly and 35xy4 – 14x4y2z + 56x3y3z3 = 7xy2(5y2 – 2x3z + 8x2yz3). ✔ 24 3A.2.2: Factoring Expressions by the Greatest Common Factor Guided Practice: Example 2, continued 25 3A.2.2: Factoring Expressions by the Greatest Common Factor