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Example Question #92 : Single Variable Algebra
A polynomial of degree 4 has as its lead term
and has rational coefficients. One of its zeroes is
; this zero has multiplicity two.
Which of the following is this polynomial?
Cannot be determined
A fourth-degree, or quartic, polynomial has four zeroes, if a zero of multiplicity is counted
times. Since its lead term is
, we know by the Factor Theorem that
where the terms are the four zeroes.
A polynomial with rational coefficients has its imaginary zeroes in conjugate pairs. Since is such a polynomial, then, since
is a zero of multiplicity 2, so is its complex conjugate
. We can set
and
, and
We can rewrite this as
or
Multiply these factors using the difference of squares pattern, then the square of a binomial pattern:
Therefore,
Multiplying:
Example Question #91 : Single Variable Algebra
Find the roots to:
This equation cannot be factored. Since this is a parabola, we can use the quadratic equation to find the roots.
Substitute the coefficients of .
The answers are:
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