Answer :
To find the possible rational zeros of the polynomial
[tex]$$
l(x)=x^3-19x^2-8x+21,
$$[/tex]
we use the Rational Root Theorem. This theorem tells us that any possible rational zero, written in lowest terms as
[tex]$$
\frac{p}{q},
$$[/tex]
must satisfy:
1. [tex]$p$[/tex] is a factor of the constant term.
2. [tex]$q$[/tex] is a factor of the leading coefficient.
Let's go through the process step by step.
1. Identify the Constant Term and the Leading Coefficient:
The constant term is the term without [tex]$x$[/tex], which here is [tex]$21$[/tex].
The leading coefficient is the coefficient of the highest power of [tex]$x$[/tex], which here is [tex]$1$[/tex].
2. Find the Factors:
- The factors of the constant term [tex]$21$[/tex] are:
[tex]$$
\pm 1,\ \pm 3,\ \pm 7,\ \pm 21.
$$[/tex]
- The factors of the leading coefficient [tex]$1$[/tex] are:
[tex]$$
\pm 1.
$$[/tex]
3. List the Possible Rational Zeros:
Since any rational zero must be of the form
[tex]$$
\frac{p}{q},
$$[/tex]
where [tex]$p$[/tex] is a factor of [tex]$21$[/tex] and [tex]$q$[/tex] is a factor of [tex]$1$[/tex], we have:
[tex]$$
\frac{\pm 1, \pm 3, \pm 7, \pm 21}{\pm 1} = \pm 1,\ \pm 3,\ \pm 7,\ \pm 21.
$$[/tex]
Thus, the possible rational zeros of the polynomial [tex]$$l(x)=x^3-19x^2-8x+21$$[/tex] are:
[tex]$$
\boxed{\pm 1,\ \pm 3,\ \pm 7,\ \pm 21.}
$$[/tex]
[tex]$$
l(x)=x^3-19x^2-8x+21,
$$[/tex]
we use the Rational Root Theorem. This theorem tells us that any possible rational zero, written in lowest terms as
[tex]$$
\frac{p}{q},
$$[/tex]
must satisfy:
1. [tex]$p$[/tex] is a factor of the constant term.
2. [tex]$q$[/tex] is a factor of the leading coefficient.
Let's go through the process step by step.
1. Identify the Constant Term and the Leading Coefficient:
The constant term is the term without [tex]$x$[/tex], which here is [tex]$21$[/tex].
The leading coefficient is the coefficient of the highest power of [tex]$x$[/tex], which here is [tex]$1$[/tex].
2. Find the Factors:
- The factors of the constant term [tex]$21$[/tex] are:
[tex]$$
\pm 1,\ \pm 3,\ \pm 7,\ \pm 21.
$$[/tex]
- The factors of the leading coefficient [tex]$1$[/tex] are:
[tex]$$
\pm 1.
$$[/tex]
3. List the Possible Rational Zeros:
Since any rational zero must be of the form
[tex]$$
\frac{p}{q},
$$[/tex]
where [tex]$p$[/tex] is a factor of [tex]$21$[/tex] and [tex]$q$[/tex] is a factor of [tex]$1$[/tex], we have:
[tex]$$
\frac{\pm 1, \pm 3, \pm 7, \pm 21}{\pm 1} = \pm 1,\ \pm 3,\ \pm 7,\ \pm 21.
$$[/tex]
Thus, the possible rational zeros of the polynomial [tex]$$l(x)=x^3-19x^2-8x+21$$[/tex] are:
[tex]$$
\boxed{\pm 1,\ \pm 3,\ \pm 7,\ \pm 21.}
$$[/tex]
