Find a polynomial function of least degree having only real coefficients, a leading coefficient of 1, and zeros of and The polynomial function is f(x) nothing. (Simplify your answer.)
Answers
Answer:
The Rational Zero Theorem tells us that if \displaystyle \frac{p}{q}
q
p
is a zero of \displaystyle f\left(x\right)f(x), then p is a factor of –1 and q is a factor of 4.
{
p
q
=
factor of constant term
factor of leading coefficient
=
factor of -1
factor of 4
The factors of –1 are \displaystyle \pm 1±1 and the factors of 4 are \displaystyle \pm 1,\pm 2±1,±2, and \displaystyle \pm 4±4. The possible values for \displaystyle \frac{p}{q}
q
p
are \displaystyle \pm 1,\pm \frac{1}{2}±1,±
2
1
, and \displaystyle \pm \frac{1}{4}±
4
1
.
These are the possible rational zeros for the function. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let’s begin with 1.
Synthetic division with 1 as the divisor and {4, 0, -3, -1} as the quotient. Solution is {4, 4, 1, 0}