Answer: The correct answer is: [D]: " (x - 5) "
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Step-by-step explanation:
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First, let's try to "factor out" the given trinomial:
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" 5x² - 5x - 100 " ;
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We can "factor out" a "5" ;
since among the three (3) terms in this trinomial,
We have: 5x² , 5x, and 100 ;
Note that not all of these 3 terms have a variable in them;
→ Consider the value; "100" .
Note that the remaining two (2) terms have the same variable, "x" ;
→ 5x² ; and 5x " ;
Note that each of these 2 (two) terms have a coefficient of "5" ;
Note that the other term, "100" ; is divisible "evenly" by "5" ;
→ that is; when the term, "100", is divided by "5" ; the result is an integer; {specially, " 100 ÷ 5 = 20 " .}.
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So: Let us being by "factoring out" a "5" ;
Given the trinomial: " 5x² - 5x - 100 " ;
→5(x² - 1x - 20) ; or: 5(x² - x - 20) ;
→ At this point, although this trinomial express is not completely
"factored out" ;
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Let us consider: Do either of these two (2) factors appear among the answer choices given:
1) " 5" ; or: 2) "(x² - x - 20)" ?? No!
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Let us continue.
We have: 5(x² - x - 20) ;
→ We can write as: " 5 (x² - 1x - 20) " ;
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Now, let see if we can further "factor out" the:
→ (x² - 1x - 20) ;
Now, note the " -20" ; and the " -1x " ;
→ What two (2) factors of: "-20" ; add up to "-1" ? ;
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Let us consider the factor of "-20" :
-20, 1 : -20 + 1 = -19 ; -19 =? -1 ? No.
20, -1 : 20 + (-1) = 20 - 1 = 19 ; 19 =? -1 ? No.
10, -2 : 10 + (-2) = 10 - 2 = 8 ; 8 =? -1 ? No.
-10, 2 : -10 + 2 = -8 ; -8 =? -1 ? No.
-4, 5 : - 4 + 5 = 5 + (-4) = 5 - 4 = 1 ; 1 =? -1 ? No.
4, -5 : 4 + (-5) = -5 + 4 = -1 ; -1 =? -1 ? Yes!
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So, we have +4 and -5 ;
(+4) * (-5) = -20 ;
(+4) + (-5) = -5 + (+4) = -5 + 4 = -1 ;
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So, we factor out:
→ (x² - 1x - 20) ;
that is: " (x² - x - 20) " ;
into:
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" (x + 4) (x - 5) " ;
and to factor out: " 5x² - 5x - 100 " ;
we bring down the "5" ; and write as:
→ " 5 (x + 4) (x - 5) " ,
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So:
We have three (3) factors when the trinomial expression is "factored out":
→ 5 ; (x + 4) ; and: (x - 5) ;
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→ Among these three factors, the only one that appears as a given answer choice is:
→ " (x - 5) " ;
→ which is our answer:
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→ Answer choice: [D]: " (x - 5) " .
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Hope this answer is helpful to you!
Best wishes!
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