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. 4x⁵ − 64x

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're given is seven P raised to the fifth power minus 112 P. Our answer choices are a seven P multiplied by the parentheses. P minus two multiplied by the parentheses. P plus two multiplied by the parentheses. P squared plus four B P multiplied by the parentheses, P minus two multiplied by the parentheses. P plus two multiplied by the parentheses. Seven P squared plus four, C seven multiplied by the parentheses. P minus two, multiplied by the parentheses. P plus two multiplied by the parentheses P squared plus four D. The given binomial is prime rewriting the expression we were given seven P raised to the fifth power minus 112 P. I recall the first thing I would like to do any time. I'm trying to factor an expression is to first check to see if there is a greatest common factor. So I want to see if there's a value that I can divide evenly out of each term starting by looking at the coefficients of 7 and -112. I wanted to check to see if seven goes evenly into 112. And when I put it in the calculator, it does. So seven is part of my greatest common factor or G C F. Next, looking at the variables I have P raised to the fifth power and P which is the same as P to the first power. This means I can factor out P to the first power from both terms. Seven P is my G C F and I'm gonna multiply that by a parentheses. And then I'm gonna do 7 p raised to the 5th power divided by 7 p. seven, divided by 7 is 1 and cancels out for our purposes. P raise the fifth power divided by P raise to the first power. We keep our base of P and we subtract our exponents. So five minus one gives me four. So I get P raised to the fourth power next negative 112 P divided by seven P, -112, divided by 7 is -16 P divided by P will cancel out. So at this step, we have 7 p multiplied by the parentheses. P raised the 4th power -16. I need to check to see if this can be factored any further. Looking at the parentheses. P raised the fourth power minus 16. I recognize this is the difference of two squares. So I'm gonna factor it as such. So seven P remains on the outside of the parentheses. And I'm opening two sets of parentheses to create two binomials. I recall that when I factor the difference of two squares, the first term in my new binomials is a result of the square root of the first term in the difference of two squares. So the square root of P raised to the fourth power is P squared. So that is the first term in each new binomial. The second term is the square root of the second term in the difference of two squares. So the square root of 16, which is four, I also recall that I need to put a minus sign in one set of the parentheses and a plus sign in the other. So after that step of factorization, we have seven P multiplied by open parentheses. P squared minus four, closed parentheses multiplied by open parentheses. P squared plus four, closed parentheses. I need to see if there's any more factorization that can be done here. I notice I have P squared minus four in one of my parentheses. P squared minus four is another example of the difference of two squares because four is a perfect square as is P squared. So we're gonna have to factor that even further. So 7 p is just getting rewritten and I'm again, gonna open two empty sets of parentheses. So I can factor P squared minus four into two binomials just like before to find the first term in each binomial, I take the square root of the first term in the difference of two squares. So the square root of P squared is P, that's the first term in each new binomial, the square of the last term. So the square of 4 is 2. That is the second term in each new binomial, we put a minus sign in one set of parentheses and a plus sign in the other. And then I need to finish rewriting what we didn't change. So the parentheses P squared plus four, we can't factor P squared plus four because that's addition and difference of two squares only works when we have a difference or subtraction. This should be as factored as our expression can get. So our final factorization is seven P multiplied by the open parentheses. P minus two closed parentheses, multiplied by open parentheses. P plus two closed parentheses, multiplied by open parentheses. P squared plus four closed parentheses. And that is answer choice. A have a nice day.

In Exercises 1–68, factor completely, or state that the polynomial is prime. 4x⁵ − 64x

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Prime Polynomials

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