Question

Which polynomial function has a leading coefficient of –3 and roots –2, Sqrt 3 and 4, all with multiplicity 1?

A) f(x)=−3(x+2)(x−√3)(x−4)



B) f(x)=(x+3)(x+2)(x−√3)(x−4))

C) f(x)=(x+3)(x+2)(x−√3)(x+√3)(x−4)


D) f(x)=−3(x+2)(x−√3)(x+√3)(x−4)

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Answer 1

The correct polynomial function is option A) f(x)=∓3(x+2)(x√−3)(x−4), which has the leading coefficient of –3 and includes factors for all roots with multiplicity 1.

Step-by-step explanation:

The student's question is asking to identify the polynomial function with a leading coefficient of –3 and has the roots −2, √3, and 4, each with multiplicity 1. A leading coefficient is the coefficient of the term with the highest power in a polynomial, in the case of a cubic polynomial like this one (with three roots), it would be the coefficient of the x3 term. Multiplicity refers to how many times a particular root is repeated within the polynomial. Given this information, we can construct the polynomial by creating factors from each root. For root −2, the corresponding factor is (x + 2), for the root √3, the factor is (x − √3), and for the root 4, it is (x − 4). Multiplying these factors together gives us the polynomial function, which should also have a leading coefficient of –3. So we need to multiply by –3 to ensure this. The correct answer is option A) f(x)=∓3(x+2)(x√−3)(x−4), which features all the required roots with the correct leading coefficient and multiplicity 1 for each root.