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Answer :
Final answer:
To determine if 3x+12 is a factor of the polynomial 3x^3+12x^2+12x+48, we can find the roots of the factor and substitute them into the original polynomial. Upon substitution, we find that x=-4 results in the polynomial equating to zero, proving that 3x+12 is indeed a factor.
Explanation:
To determine if 3x+12 is a factor of 3x^3+12x^2+12x+48, we can use polynomial division or check for a root that would make 3x+12 equal to zero. If the polynomial can be divided evenly by 3x+12 (meaning without a remainder), then it is a factor.
First, we find the root of the potential factor 3x+12 by setting it equal to zero:
3x+12=0
3x=-12
x=-4
Next, we can substitute this value into the original polynomial to check if it results in zero:
3(-4)^3+12(-4)^2+12(-4)+48
3(-64)+12(16)-48+48
-192+192-48+48 = 0.
Since substituting x=-4 into the polynomial equals zero, 3x+12 is indeed a factor of 3x^3+12x^2+12x+48.
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