Final answer:
In the polynomial function given, the multiplicity of the factor (x²+4) is 2 and the factor (x-4) is 3. Multiplicity indicates how the graph behaves at the roots. Exponents affect all terms within parentheses and raising to a power involves multiplying the base by itself as many times as indicated by the exponent.
Step-by-step explanation:
The student's question revolves around the function f(x) = 2(x²+4)²(x-4)³. When discussing the function's multiplicity, we are focusing on the exponents of the factors in the function. Multiplicity refers to the number of times a particular factor appears as a root in the polynomial.
For this function, the factor (x²+4) has a multiplicity of 2, and the factor (x-4) has a multiplicity of 3. The concept of multiplicity is key in understanding the behavior of the graph of the polynomial near its roots.
To understand how operations such as completing the square work, remember that when you have an expression like (2x)² = 4.0(1 − x)², you may take the square root of both sides to simplify. Applying integer powers, like in the expression (27x3)(4x²), involves knowing that the power affects everything inside the parentheses.
In the case of raising a number to a power, such as 4³, this is equivalent to multiplying the number by itself the number of times indicated by the exponent, a fundamental concept for understanding how to manipulate arithmetic expressions involving exponents.