multiply and simplify.
3b^2-27/12b^2-300 x 48b+240/4b-12
To multiply and simplify the given expression:
Step 1: Simplify each fraction separately:
- Simplify the numerator of the first fraction: 3b^2 - 27
- Start by factoring out the greatest common factor (GCF) from both terms, which is 3:
3b^2 - 27 = 3(b^2 - 9)
- Notice that (b^2 - 9) is a difference of squares, so it can be further factored:
b^2 - 9 = (b - 3)(b + 3)
- Therefore, the numerator becomes: 3(b - 3)(b + 3)
- Simplify the denominator of the first fraction: 12b^2 - 300
- Start by factoring out the greatest common factor (GCF) from both terms, which is 12:
12b^2 - 300 = 12(b^2 - 25)
- Notice that (b^2 - 25) is a difference of squares, so it can be further factored:
b^2 - 25 = (b - 5)(b + 5)
- Therefore, the denominator becomes: 12(b - 5)(b + 5)
- Simplify the numerator of the second fraction: 48b + 240
- Start by factoring out the greatest common factor (GCF) from both terms, which is 48:
48b + 240 = 48(b + 5)
- Simplify the denominator of the second fraction: 4b - 12
- Start by factoring out the greatest common factor (GCF) from both terms, which is 4:
4b - 12 = 4(b - 3)
Step 2: Multiply the two fractions:
- Multiply the numerators together: [3(b - 3)(b + 3)] * [48(b + 5)]
- Combine the like terms: 3 * 48 = 144
- Multiply the binomials: (b - 3)(b + 3) * (b + 5)
- Apply the FOIL method to multiply:
(b - 3)(b + 3) = b^2 - 3b + 3b - 9 = b^2 - 9
- Multiply (b^2 - 9) * (b + 5):
- Combine like terms: (b^2 - 9)(b + 5) = b^3 + 5b^2 - 9b - 45
- Multiply the denominators together: [12(b - 5)(b + 5)] * [4(b - 3)]
- Multiply the binomials: (b - 5)(b + 5) = b^2 - 25
- Multiply (b^2 - 25) * (b - 3):
- Combine like terms: (b^2 - 25)(b - 3) = b^3 - 3b^2 - 25b + 75
Step 3: Combine the results obtained in Step 2:
- The product of the two fractions is:
num_result = 144 * (b^3 + 5b^2 - 9b - 45)
den_result = 12 * 4 * (b^3 - 3b^2 - 25b + 75)
Step 4: Simplify the results by canceling out common factors, if any, from the numerator and denominator:
- Simplify the numerator: Multiply 144 with the polynomial in (b):
num_result = 144b^3 + 720b^2 - 1296b - 6480
- Simplify the denominator: Multiply 12 * 4 and cancel out any common factors:
den_result = 48(b^3 - 3b^2 - 25b + 75)
Step 5: Simplify the final expression:
- Now we have the simplified expression:
(144b^3 + 720b^2 - 1296b - 6480) / (48b^3 - 144b^2 - 1200b + 360)
And that's the final simplified expression.