In Exercises 1–30, factor each trinomial, or state that the trino...

Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.y² − 12y + 20

Accepted Answer

Mhm. Hello everyone. We've been asked to perform a factorization in the given expression or categorize it as a prime trinomial. Also, we are going to perform a check using the foil method. The trinomial we are given is Z squared minus 20 Z plus 75. Our answer choices are a, the parentheses, Z -5 multiplied by the parentheses. Z -15 B, the parentheses, Z -3 multiplied by the parentheses. Z -25 C, the parentheses, Z -1 multiplied by the parentheses. Z -75 D, the parentheses Z plus five multiplied by the parentheses. Z plus 15, rewriting my original trinomial. I have Z squared minus 20 Z plus 75. And I recall when I factor a trinomial, it typically results in two binomials. So I'm gonna open two sets of parentheses next to each other. They're currently empty, but we will fill them using information from the trinomial. First thing I'm gonna do is fill in the first term in each set of parentheses and that comes from two values that will multiply to the first term in the trinomial. The first term in our trinomial is Z squared. So what will multiply that is Z and Z so Z multiplied by Z would give me Z squared. So that is the first term in each binomial. Next, I need two values that not only multiply to a positive 75, but add to a negative 20. Before I start trying to guess and check values. I recall that if I need things to multiply to a positive but add to a negative, I'm gonna need two negative values. So instead of having to keep track of that, while I check multiple multiplication, I could put my minus signs in my parentheses right now. So both sets of parentheses now have Z and A minus sign from there. I can write down all the combinations I can think of that multiply to 75. So one in 75, 3 and 25, five and 15 and I'm gonna check those three and if I don't find one that works, I will see if there's more combinations, but we're gonna stop there for now. So we're gonna check one multiplied by 75. We know will give us 75. But does it add to 20? No, it does not. 3 and 25. Again, we know that multiplies to 75 but does it add to 20? No, it does not. Five and 15. We again know that multiplies 75 we need to check, does that add to 20? It does. So 5 and 15 are the last values in my binomial. So I have the parentheses, Z -5 multiplied by the parentheses Z -15. This is likely our factorization but we want to check it first using the foil method. So we are going to check recall that foil stands for first outer, inner and last. So first would be the first term in the binomials. So Z multiplied by Z it results in Z squared outer. So the outer terms from the binomial Z multiplied by negative 15, results in negative 15 Z dinner, negative five multiplied by Z gives me negative five. Z last is the last term in each binomial. So negative five multiplied by -15 is a positive 75. Currently, I have Z squared minus 15, Z minus five, Z plus 75. I have like terms in my negative 15 Z and negative five Z I will be combining them as I rewrite the expression. So Z squared minus 20 Z plus 75. This does match my original trinomial which means our factorization is correct. So parentheses, Z -5 multiplied by the parentheses Z -15 is our solution and it is answer choice. A have a nice day.

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In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.y² − 12y + 20

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