Q:

Which of the binomials below is a factor of this trinomial?5х2 - 5x - 100А. х-4ОВ. x+ 10ос. x+5OD. х- 5

Accepted Solution

A:
Answer:  The correct answer is:   [D]:  " (x - 5) "__________________________________________________Step-by-step explanation:__________________________________________________First, let's try to "factor out" the given trinomial:__________________________________________________    "  5x² -  5x  - 100 " ; __________________________________________________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 " .}.__________________________________________________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" ; __________________________________________________Let us consider:  Do either of these two (2) factors appear among the answer choices given:           1)  " 5" ;  or:  2)   "(x² - x - 20)" ??  No!__________________________________________________Let us continue.We have:  5(x² - x - 20) ;          →  We can write as:  " 5 (x² - 1x - 20) "  ; __________________________________________________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" ?  ; __________________________________________________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!__________________________________________________So,  we have +4  and -5 ;     (+4) * (-5) = -20 ;   (+4) + (-5)  =  -5 + (+4) =  -5 + 4 = -1 ; __________________________________________________So, we factor out:  →  (x² - 1x - 20) ;that is:  " (x² - x - 20) " ; into:__________________________________________________" (x + 4) (x - 5) " ; and to factor out:  " 5x² - 5x - 100 " ; we bring down the "5" ;  and write as:  →  " 5 (x + 4) (x - 5) " ,__________________________________________________So:We have three (3) factors when the trinomial expression is "factored out":    →  5 ;  (x + 4) ;  and:  (x - 5) ; __________________________________________________  → Among these three factors, the only one that appears as a given answer choice is:    →  " (x - 5) " ;  →  which is our answer:__________________________________________________  →   Answer choice:  [D]:  " (x - 5) " .__________________________________________________Hope this answer is helpful to you!       Best wishes!__________________________________________________