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. x³ − 4x² − 9x + 36

Accepted Answer

Hello, everyone. We've been asked to perform a complete factorization of the given expression or categorize it as the prime polynomial. The expression we were given is M cubed minus three M squared minus 16, M plus 48. Our answer choices are a open parentheses. M minus three closed parentheses multiplied by open parentheses. M minus four closed parentheses multiplied by open parentheses. M plus four closed parentheses. B open parentheses, M minus three closed parentheses, multiplied by open parentheses. M plus four closed parentheses. Squared, C open parentheses, M -3 closed parentheses, multiplied by open parentheses. M -4 closed parentheses. Squared D, the given polynomial is prime rewriting the polynomial we have M squared. I'm sorry. M cubed minus three M squared minus 16 M plus 48. I recall that when I'm asked to factor something, I want to look for greatest common factor first. And in this case, there is no greatest common factor between all four terms. Next. I recognize that I have four terms. And I recall that typically when I have four terms, I end up factoring by grouping. So that's what I'm gonna do here. I'm gonna factor by grouping I'm gonna group together the first two terms in parentheses. So M cubed minus three M squared are now in a set of parentheses. And then I'm gonna group together the last two terms. So negative 16 M plus 48 are now also in a set of parentheses from there. I want to factor a greatest common factor out of each set of parentheses. So looking at the first set of parentheses, M cubed minus three M squared, I wanna see what is the largest thing I could take out of both terms. And that appears to be an M squared. So I'm gonna write down M squared open a set of parentheses and divide each term from my first set of parentheses by M squared to find out what's in my new parentheses. So M cubed divided by M squared will result in M negative three M squared divided by M squared will leave me with a negative three closed parentheses. Now looking at my second set of parentheses from my original expression, negative 16 M plus 48. I'm looking again for greatest common factor and I'm looking at the coefficient and the constant and I am positive that 16 goes into 48. But since there's a negative in front of my 16, I'm actually gonna factor this out as negative 16 for my greatest common factor. Then open parentheses and I'm gonna divide both terms by -16. So negative 16 M divided by negative 16 leaves us with M 48 divided by -16 leaves us with a negative three closed parentheses. So at this step, I have M squared multiplied by open parentheses. M -3 closed parentheses -16, multiplied by open parentheses. M -3 closed parentheses. Now I've done the first step in factoring and I've created a binomial G C F. So the parentheses M -3 appear in both terms. So that's a binomial, greatest common factor. And I could pull that to the front as open parentheses. M -3, closed parentheses. Then I'm gonna multiply that by another set of parentheses. And in those parentheses, I'm gonna place what I had as my G C F in my previous step. So M squared -16, closed parentheses. So at this step in factorization, I have open parentheses, M minus three, closed parentheses multiplied by open parentheses. M squared minus 16, closed parentheses. I wanna see if this can be factored any further and looking at my second set of parentheses. M squared minus 16. I recognize that that is the difference of two squares. So I'm gonna factor this further. The parentheses M -3 are not going to change. I just rewrote them and then I'm going to open two sets of parentheses next to each other and leave them empty for this second. I'm gonna fill these with two binomials that will result from factoring the difference of two squares. I recall that when I factor the difference of two squares, the first term in each new binomial comes from the square root of the first term of the difference of two squares. So the square root of M squared is M and that is the first term in each of my new binomials. The second term comes from the square root of the second term in the difference of two squares. So in this case, the square root of 16, which is four. So that is the second term in each binomial. And lastly, I need to put opposite signs in between those terms. So one needs a minus sign and one needs a plus sign. And once I do that, my factorization is open parentheses, M minus three closed parentheses, multiplied by open parentheses, M minus four, closed parentheses, multiplied by open parentheses. M plus four closed parentheses. This is my factorization and it is answer choice. A have a nice day.

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In Exercises 1–68, factor completely, or state that the polynomial is prime. x³ − 4x² − 9x + 36

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