In Exercises 1–68, factor completely, or state that the polynomia...

Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 4a²b − 2ab − 30b

Accepted Answer

Hello, everyone. We have been asked to perform a complete factorization of the given expression or categorize it as the prime polynomial. The expression we're given is 12 M squared N minus 159 M N minus 315 N. Our answer choices are a three N multiplied by the parentheses. Four M plus seven multiplied by the parentheses. M minus 15 B three multiplied by the parentheses. Four M plus seven multiplied by the parentheses. M minus 15 C three N multiplied by the parentheses. Four M minus seven multiplied by the parentheses M plus 15 D. The given trinomial is prime. I recall that as with all factorization, I first wanna look for a greatest common factor. So I'm gonna rewrite my original expression. 12 M squared N minus 159 M N minus 315 N. And now I want to see if there's a value that I can divide evenly out of each term. I'm gonna start off by looking at my coefficients 12, -159 and -315. I know that 12 is not going to go evenly into 159 or 315 because the even number will not divide evenly into odd numbers. So I'm thinking of what odd factor is in 12. And I come up with three, once I check this three does go evenly into each coefficient. So three is part of my greatest common factor. Next, I look at the variables. Each term does not have an M value, but each term does have an N value. And as such, I can factor N out of each term. So three N is my greatest common factor. I'm gonna multiply that by open parentheses. And I'm gonna divide each term from my original expression by my greatest common factor of 3 n. So 12 M squared N divided by three N result in four M squared. Next negative 159 M N divided by three N give me negative M and negative 315 N divided by three N would be negative 105 closed parentheses. So at this step, I have three N multiplied by the parentheses, four M squared minus 53 M minus 105 closed parentheses. I can factor that trinomial further. And so I'm gonna open a new set of parentheses. So I have three N multiplied by and I'm opening two empty sets of parentheses. I recall that to factor a trinomial, it will factor into two binomials. So to factor this into two binomials, I need the first term of my new binomials to multiply to the first term of my trinomial. So my choices are to multiply the four M squared, four M multiplied by M or two M multiplied by two M. I'm going to go with four M multiplied by M. And then I need two values that will multiply to a -105. And in that, I need to make sure that I can check my inner and outer terms to get a negative 53 M. So values that will multiply the -105 are gonna be, there's not a lot of them, one and 105, but that's not gonna work well. Um 7 and 15 sound good since it looks like I want my larger value to be negative. I'm gonna put the negative sign with the 15 and multiply that by the four and the positive sign with the seven. So I end up with the factorization three N multiplied by open parentheses, four M plus seven, closed parentheses, multiplied by open parentheses M minus 15, closed parentheses. Before I say this is right, I'm gonna check this. So to check it, I'm gonna multiply my inner and outer terms and make sure that comes out to a negative 53 M. So inner terms would be seven times M to seven M. Outer terms are negative 15, multiplied by positive four M. So that would be a negative 60 m and when I add seven M plus negative 60 M. That does give me negative 53 M. Since my middle term checks out, I know my factorization is correct. So my answer is three N multiplied by open parentheses. Four M plus seven closed parentheses multiplied by open parentheses. M minus 15 closed parentheses. And that's answer choice. A have a nice day.

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In Exercises 1–68, factor completely, or state that the polynomial is prime. 4a²b − 2ab − 30b

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