find the rational expression R whose numerator and denominator have degree 2 and leading coefficients of 1 such that (x^2 + 3x-10)/(x^2 -8x+15) * R = (x-2)/(x-3)
x^2+3x-10 = (x+5)(x-2)
x^2-8x+15 = (x-5)(x-3)
So, you want
(x+5)(x-2) / (x-5)(x-3) * R = (x-2)/(x-3)
So, you have
R = (x-2)/(x-3) * (x-5)(x-3) / (x+5)(x-2)
R = (x-5)/(x+5)
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To find the rational expression R, let's first simplify the given equation:
(x^2 + 3x - 10)/(x^2 - 8x + 15) * R = (x - 2)/(x - 3)
To simplify further, let's factor the given quadratic expressions:
(x^2 + 3x - 10) = (x - 2)(x + 5)
(x^2 - 8x + 15) = (x - 3)(x - 5)
Substituting these into the equation, we get:
[(x - 2)(x + 5)] / [(x - 3)(x - 5)] * R = (x - 2)/(x - 3)
Now, we can cancel out the common factors:
[(x - 2)(x + 5)] / [(x - 3)(x - 5)] * R = (x - 2)/(x - 3)
Canceling out the common factor (x - 2) on both sides, we get:
(x + 5) / [(x - 3)(x - 5)] * R = 1/(x - 3)
To solve for R, we can cross-multiply:
(x + 5) * R = (x - 3) * [(x - 3)(x - 5)]
Expanding the right side:
(x + 5) * R = (x - 3) * (x^2 - 8x + 15)
Multiplying through:
(x + 5) * R = x^3 - 2x^2 - 7x + 45
Now, we can divide both sides by (x + 5) to isolate R:
R = (x^3 - 2x^2 - 7x + 45) / (x + 5)
Therefore, the rational expression R is:
R = (x^3 - 2x^2 - 7x + 45) / (x + 5)
To find the rational expression R, we can start by multiplying both sides of the equation by the denominator of the left-hand side:
(x^2 + 3x - 10) * R = (x - 2)(x^2 - 8x + 15)/(x - 3)
Next, we can expand both sides of the equation:
(x^2 + 3x - 10) * R = (x - 2)(x^2 - 8x + 15)/(x - 3)
--> x^2R + 3xR - 10R = (x - 2)(x^2 - 8x + 15)/(x - 3)
Now, let's expand the numerator on the right-hand side:
x^2R + 3xR - 10R = (x - 2)(x^2 - 8x + 15)/(x - 3)
--> x^2R + 3xR - 10R = (x^3 - 8x^2 + 15x - 2x^2 + 16x - 30)/(x - 3)
Simplify the numerator:
x^2R + 3xR - 10R = (x^3 - 10x^2 + 31x - 30)/(x - 3)
Since the numerator and denominator of R both have a degree of 2 and leading coefficients of 1, we can express R as:
R = (x^3 - 10x^2 + 31x - 30)/(x^2 + 3x - 10)
Therefore, the rational expression R is:
R = (x^3 - 10x^2 + 31x - 30)/(x^2 + 3x - 10)