In Exercises 23–48, factor completely, or state that the polynomi...

Question

In Exercises 23–48, factor completely, or state that the polynomial is prime.50 - 2y²

Accepted Answer

Hello, everyone. We're asked to completely factorize the following polynomial expression. If possible. Otherwise declare if the expression is a prime polynomial, the expression we're given is 75 -3 p squared. Our answer choices are a negative three multiplied by the parentheses. Five minus P multiplied by the parentheses. Five plus P B negative three multiplied by the parentheses. P minus five multiplied by the parentheses. P plus five C negative three multiplied by the parentheses P squared minus D. This polynomial is prime rewriting my original expression 75 minus three P squared. I'm going to first look to see if there's a greatest common factor or a value that I could take evenly out of both terms. I look at the coefficient and the constant. And I noticed that I definitely could divide three out of both three and 75. So that's gonna be part of my greatest common factor. I like to rewrite these so that my variable comes before my constant. And I can do that if I factor a negative three out instead of a positive three. So when I factor in negative three out, I have negative three multiplied by parentheses. And then 75 divided by negative three is negative negative three P squared, divided by negative three would be plus P squared closed parentheses. And this can be rewritten as negative three, open parentheses. P squared minus 25 closed parentheses. Now this does match one of our answer choices. However, this is not completely factored yet. So we need to do one more step within the parentheses. I recall that that is the difference of two squares. I have P squared and 25 is a perfect square. So I'm going to rewrite my negative three G C F. Then I'm opening two sets of parentheses next to each other. So I can factor P squared minus 25. I recall that when I factor the difference of two squares, the first term in my new binomial are from the first term in my difference of two squares. I take the square root of that first term. So P squared, the square root of it is P. So P is the first term in each parentheses. The last term in each parentheses comes from the square root of 25. So five and I recall that the signs have to be opposite. One will be a minus sign, one will be a plus sign. So my factorization is negative three multiplied by the parentheses. P minus five multiplied by the parentheses P plus five. And that matches answer choice B have a nice day.

In Exercises 23–48, factor completely, or state that the polynomial is prime.50 - 2y²

Key Concepts

Factoring Polynomials

Difference of Squares

Prime Polynomials

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