Find the Domain (30x^3-63x^2+45x)/(4x^4-8x^3+7x^2-12x+9)

Problem

(30*x3−63*x2+45*x)/(4*x4−8*x3+7*x2−12*x+9)

Solution

1. Identify the condition for the domain of a rational function, which is that the denominator must not be equal to zero.

4*x4−8*x3+7*x2−12*x+9≠0

2. Factor the denominator by grouping or testing for rational roots. Testing x=3/2 using the Rational Root Theorem:

4*(3/2)4−8*(3/2)3+7*(3/2)2−12*(3/2)+9=81/4−27+63/4−18+9=0

3. Perform synthetic division or polynomial long division to find that (2*x−3) is a factor. Dividing the polynomial by (2*x−3)2 reveals:

4*x4−8*x3+7*x2−12*x+9=(2*x−3)2*(x2+1)

4. Set each factor to zero to find the excluded values.

(2*x−3)2=0⇒x=3/2

x2+1=0⇒x2=−1

5. Determine that x2+1=0 has no real solutions, so the only real value that makes the denominator zero is x=3/2

6. State the domain in interval notation by excluding x=3/2 from the set of all real numbers.

Final Answer

Domain:(−∞,3/2)∪(3/2,∞)

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