The number of roots remaining of the polynomial function f(x) = x³ − 7x − 6, with two roots -2, and 3 already given is 1. The nature of the root will be real.
What are polynomial functions?
A polynomial function is a function (say f(x)), which is defined over a polynomial expression in x. It is of the form,
f(x) = a₀ + a₁x + a₂x² + a₃x³ + ... + aₙxⁿ, where a₀, a₁, a₂, ..., aₙ are constants, x is a variable, and n ≥ 0.
Degree of the polynomial function = n, the highest power of x.
What is the fundamental theorem of algebra?
The fundamental theorem of algebra is that the number of roots or solutions of a polynomial function = The degree of the polynomial function.
What is the complex conjugate theorem?
According to the complex conjugate theorem, if a polynomial function has complex roots, they will always exist in conjugate pairs, that is, if one root is of the form a + bi, the other root will be a - bi.
How will we determine the question?
We are given a polynomial function f(x) = x³ - 7x - 6. Two roots of the equation are given as -2, and 3.
The degree of the equation = 3, so by the fundamental theorem of algebra number of roots = 3.
2 roots are given, so the number of roots remaining = 1.
Since none of the given roots are complex, the third root can not be complex, as complex roots always exist in conjugate pairs, coming from the complex conjugate theorem. So, the remaining root will be real in nature.
Learn more about the fundamental theorem of algebra and the complex conjugate theorem at
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