How do I factor out the greatest common monomial factor?
1) 9m^7 - 3m^2
2) 12a^5 + 8a
3) I need to describe and correct the error in facotring out the greatest common monomial factor of 18x^8 - 9x^4 - 6x^3
18x^8 - 9x^4 - 6x^3 = 3x(6x^7 - 3x^3 - 2x^2) is INCORRECT
1. 3m^2(3m^5-1).
2. 4a(3a^4+2).
3. 18 = 3*3*2.
9 = 3*3.
6 = 3*2.
GCF = 3x^2.
Note: The lowest power of x was used.
To factor out the greatest common monomial factor, you need to identify the highest power of each variable that appears in each term and then find the largest common factor of those powers.
1) The highest power of "m" in the first term is 7, and the highest power in the second term is 2. The largest common factor of these powers is "m^2".
Therefore, to factor out the greatest common monomial factor from 9m^7 - 3m^2, you can rewrite it as:
9m^7 - 3m^2 = 3m^2(3m^5 - 1).
2) The highest power of "a" in the first term is 5, and the highest power in the second term is 1. The largest common factor of these powers is "a".
Therefore, to factor out the greatest common monomial factor from 12a^5 + 8a, you can rewrite it as:
12a^5 + 8a = 4a(3a^4 + 2).
3) The error in the given equation is that "18x^8 - 9x^4 - 6x^3" is factored as "3x(6x^7 - 3x^3 - 2x^2)", which is incorrect.
To correctly factor out the greatest common monomial factor from 18x^8 - 9x^4 - 6x^3, follow these steps:
Step 1: Identify the highest power of "x" in each term:
- The highest power in the first term is 8.
- The highest power in the second term is 4.
- The highest power in the third term is 3.
Step 2: Find the largest common factor of these powers:
The largest common factor of 8, 4, and 3 is 1, as there are no powers that are common to all three terms.
Therefore, the correct way to factor out the greatest common monomial factor from 18x^8 - 9x^4 - 6x^3 is to leave it as is, since there is no common factor to all the terms.
To factor out the greatest common monomial factor, you need to find the largest factor that divides into all the terms in the expression.
Let's go through each problem one by one:
1) 9m^7 - 3m^2
In this case, the greatest common monomial factor is 3m^2. To factor it out, you divide every term in the expression by 3m^2:
9m^7 / (3m^2) = 3m^(7-2) = 3m^5
-3m^2 / (3m^2) = -1
Therefore, the factored form is: 3m^2(3m^5 - 1)
2) 12a^5 + 8a
Here, the greatest common monomial factor is 4a. Divide each term by 4a:
12a^5 / (4a) = 3a^(5-1) = 3a^4
8a / (4a) = 2
The factored form is: 4a(3a^4 + 2)
3) The error in factoring out the greatest common monomial factor of 18x^8 - 9x^4 - 6x^3 is in the solution provided.
The incorrect factored form is: 3x(6x^7 - 3x^3 - 2x^2)
To correct it, we need to find the actual greatest common monomial factor. Let's look at the coefficients first. The coefficients of the terms are 18, -9, and -6. The greatest common factor of these coefficients is 3.
Now, let's consider the variable part. The exponent of x in the first term is 8, in the second term is 4, and in the third term is 3. The greatest exponent is 8.
Therefore, the greatest common monomial factor is 3x^8. Divide each term by 3x^8:
18x^8 / (3x^8) = 6
-9x^4 / (3x^8) = -3x^(4-8) = -3x^(-4)
-6x^3 / (3x^8) = -2x^(3-8) = -2x^(-5)
The correct factored form is: 3x^8(6 - 3x^(-4) - 2x^(-5))
Remember, when dividing exponents, subtracting the exponents is equivalent to dividing the variables.
I hope that helps! Let me know if you have any more questions.