Can someone help me understand this concept?
If (3x + 9) = (ax + b), then a = 3 and b = 9
If (3x + 9)/2 = (ax + b)/c, then why can’t a = 3, b = 9, and c = 2?
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Reference GregMat Question:
Two polynomials are equal iff they have the same degree and the coefficients of corresponding terms match
Are you saying that the below equation won’t work because 1/c or c-1 invalidates the polynomial equation condition?
@cylverixxx
Do let me know if I need to clarify more
Rephrased question:
If (3x + 9)/2 = (ax + b)/c, will it always be true that a = 3, b = 9, and c = 2? (Given that c != 0)
From what i said prior, you require the following:
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\frac ac = \frac 32
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\frac bc = \frac 92
As you can imagine, there are many such (a,b,c) that satisfy both conditions.
For example, (a,b,c) = (6,18,4) would also work, so (a,b,c) = (3, 9, 2) is certainly not a necessary condition.
@cylverixxx
Thank you for explaining it in detail. I understood it now.
Just one last question:
Can a and b have multiple values in the following case :
If (3x + 9) = (ax + b), then a = 3 and b = 9, because there are no other values that a and c can ake
Correct me if I am wrong, but I think they can’t have any other values because there are no ratios/proportions being formed here.
(3x + 9) - (ax + b) = (3 - a)x + ( 9 - b) = 0
Thus, we require that 3 - a =0 and 9 - b = 0 if this equality between two polynomials is to hold for all x.
Alternatively, you could also match the degree of both sides of the equation:
\operatorname{deg}((3 - a)x + (9 - b)) = \operatorname{deg}(0)
If a \neq 3 then you’ll have that the degree of the left side is 1, while the degree of the right side (the zero polynomial) is -\infty, which clearly doesn’t match. The same rationale also necessitates that b = 9 because nonzero constants have a different degree than the zero polynomial.
In essence, you just need to match the coefficients of the corresponding terms if you want to establish equality between polynomials.
It’s always about matching the corresponding terms , so I would say the part about “ratios/proportions” is irrelevant.
Both sides should represent the same line, so ax + b must have a slope of 3 and y-intercept of 9. I think that’s a pretty straightforward way to look at it for this particular example.
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@cylverixxx
Thank you for taking the time to explain it in so much detail. It helped understand the concept better
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