Task 1: Determine if the given equation has 1 solution, NO solution, or INFINITELY many solutions. Explain your
2 answers:
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Answer: " 2x (2x - 1) (x + 1) " .
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Step-by-step explanation:
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Given:
f(x) = 9x³ + 2x² − 5x + 4 ;
g(x) = 5x³ − 7x + 4 ;
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What is: f(x) − g(x) ?
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Plug in: " 9x³ + 2x² − 5x + 4 " for: " f(x) " ;
and: " (5x³ − 7x + 4) " ; for: "g(x)" ;
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→ " f(x) − g(x) =
" 9x³ + 2x² − 5x + 4 − (5x³ − 7x + 4) " .
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Rewrite this expression as:
→ " 9x³ + 2x² − 5x + 4 − 1(5x³ − 7x + 4) " .
→ {since: " 1 " ; multiplied by "any value" ; is equal to that same value.}.
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Now, let us example the following portion of the expression:
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" − 1(5x³ − 7x + 4) "
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Note the "distributive property" of multiplication:
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→ a(b + c) = ab + ac ;
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Likewise:
→ a(b + c + d) = ab + ac + ad .
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As such:
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→ " − 1(5x³ − 7x + 4) " ;
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= (-1 * 5x³) + (-1 * 7x) + (-1 * 4) ;
= - 5x³ + (-7x) + (-4) ;
= - 5x³ − 7x − 4 ;
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Now, add the "beginning portion of the expression" ; that is:
" f(x) " ; to the expression ; which is:
→ 9x³ + 2x² − 5x + 4 ;
→ as follows:
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→ 9x³ + 2x² − 5x + 4 − 5x³ − 7x − 4 ;
→ {Note that the: " - " sign; that is;
the "negative sign", in the term: " -5x³ " ;
becomes a: " − " sign; that is; a "minus sign" .}.
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Now, combine the "like terms" of this expression; as follows:
+ 9x³ − 5x³ = + 4x³ ;
− 5x − 7x = − 2x ;
+ 4 − 4 = 0 ;
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and we have:
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→ " 4x³ + 2x² − 2x ".
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Now, to write this answer in "factored form" :
Note that among all 3 (three) terms in this expression, each term has a factor of "2" . The lowest coefficient among these 3 (three) terms is "2" ; so we can "factor out" a "2".
Also, each of the 3 (three) terms in this fraction is a coefficient to a variable. That variable takes the form of "x". The term in this expression with the variable, "x"; with the lowest degree has the variable: "x" (i.e. "x¹ = x" ) ; so we can "factor out a "2x" (rather than just the number, "2".).
So, by factoring out a "2x" ; take the first term [among the 3 (three) terms in the expression] —which is: "4x³ " .
2x * (?) = 4x³ ? ;'
↔ ? ;
→ 4/2 = 2 ;
;
As such: 2x * (2x²) = 4x³ ;
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Now, by factoring out a "2x" ; take the second term [among the 3 (three) terms in the expression] — which is: "2x² " .
2x * (?) = 2x² ? ;
↔ ?
→ 2/2 = 1 ;
→ ;
As such: 2x * (x) = 2x²
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Now, by factoring out a "2x" ; take the third term [among the 3 (three) terms in the expression] — which is: " − 2x " .
2x * (?) = - 2x ;
↔ -1 ;
As such: 2x * (-1) = − 2x .
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So:
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Given the simplified expression:
→ " 4x³ + 2x² − 2x " ;
We can "factor out' a: " 2x " ; and write the this answer is: "factored form" ; as:
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"2x (2x² + x − 1 ) . "
Now, we can further factor the:
" (2x² + x − 1) " ; portion;
Note: "(2x² + x - 1)" =
2x² + 2x - 1x -1 = (2x -1) + x (2x - 1 ) =
(2x - 1) ( x + 1)
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Now, bring down the "2x" ; and write the Full "factored form" ; as follows:
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→ " 2x (2x - 1) (x + 1) " .
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Hope this helps!
Wishing you the best!
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