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16 ## COMMON FACTORTO FACTOR A NUMBER or an expression, means to write it as multiplication, that is, as a product of factors. Example 1. Factor 30.
If we begin 30 = 5 Problem 1. Factor 50. To see the answer, pass your mouse over the colored area.
50 = 2 Factoring, then, is the reverse of multiplying. When we multiply, we write 2( But if we switch sides and write 2 then we have factored 2 In this sum 2
Problem 2. Factor 3
3
Problem 3. Rewrite each of the following as the product of 2 For example, 10
Example 2. Factor 10
10 You can always check factoring by multiplying the right-hand side. It should produce the left-hand side. Also, the sum on the left has three terms. Therefore the sum in parentheses must also have three terms -- and it should have no common factors. Problem 4. Factor each sum. Pick out the common factor. Check your answer.
Problem 5. Factor each sum. a) 2 + 6 + 10 + 14 + 18 = 2(1 + 3 + 5 + 7 + 9) b) 30 + 45 + 60 + 75 = 15(2 + 3 + 4 + 5) Again, the number of terms in parentheses must equal the number of terms on the left . And the terms in parentheses should have no common factors. A monomial in 5 We say that the number 6 is a monomial in 6 = 6 A polynomial in 5 When we write a polynomial, the style is to begin with the highest exponent and go to the lowest. 4, 3, 2, 1. (For a more complete definition of a polynomial, see Topic 6 of Precalculus.) The degree of a polynomial is the highest exponent. The polynomial above is of the 4th degree. The constant term is the term in which the variable does not appear. In other words, it is the number at the end. In the example above, the constant term is −2. (We call it the constant term because even when the value of the variable changes, the value of the constant term does not change. It is constant.) A quadratic is a polynomial of the 2nd degree. Problem 6. Describe each polynomial in terms of the variable it is "in," and say its degree. a) b) 3 c) d) e) 4 Factoring polynomials If every term is a power of
then the
For, lower powers are factors of higher powers .
The lowest power, Once more, to say that we have
-- means that we will obtain that polynomial if we The student should confirm that. Problem 7. Factor these polynomials. Pick out the highest common factor. (How can you check your factoring? By multiplying!) a) b) 5 c) d) 6 e) 2 f) 3 Problem 8. Factor each polynomial. Pick out the highest common numerical factor and the highest common literal factor. a) 12 There is no common literal factor. The sum in parentheses has no common factors. b) 16 c) 36 d) 8 e) 16 f) 20 g) 18 h) 12
Example 3. Factor
If you multiply the right-hand side, you will obtain the left-hand side. Problem 9. Factor. a) 3 b) 2 c) d) 8 e) f) Next Lesson: Factoring trinomials Please make a donation to keep TheMathPage online. Copyright © 2022 Lawrence Spector Questions or comments? E-mail: [email protected] |