Chapter 2, Problem 53RE

Solution

a.

To find the : The real zeros of a the function . $f\left(x\right)=x^4-10x^3+23x^2$ .

The real zeros are $x=0,x=\left(5+\sqrt{2}\right),x=5-\sqrt{2}$ .

Given information:

The given function is $f\left(x\right)=x^4-10x^3+23x^2$ .

Concept used:

Factor theorem: A polynomial function has a factor $x-k$ if only if $f\left(x\right)=0$

Rational zero theorem: Suppose f is a function of degree $n\ge 1$ of the form

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{......}+a_0$ with every coefficient an integer and $a_0\ne 0.$ If $x=\frac{p}{q}$ is a rational zero of $f$ where $p$ and $q$ have no common integers other than $\pm 1$ then $p$ is an integer factor of the constant $a_0$ and $q$ is an integer factor of the leading coefficient $a_0$ .

Calculations:

The function

  $f\left(x\right)=x^4-10x^3+23x^2$ .Rewrite the function given as $x^2\left(x^2-10x+23\right)$ .

Factor $\left(x^2-10x+23\right)$ as $$

  $\left(x-\left(5+\sqrt{2}\right)\right)\left(x-\left(5-\sqrt{2}\right)\right)$ by using the quadratic formula

Because of polynomial factor theorem the factor are $x=0,\left(x-\left(5+\sqrt{2}\right)\right),\left(x-\left(5-\sqrt{2}\right)\right)$

Hence the zeros are $x=0,x=\left(5+\sqrt{2}\right),x=5-\sqrt{2}$

b.

To identify : the zeros are rational, irrational or complex zeros of the function . $f\left(x\right)=x^4-10x^3+23x^2$ and the zeros are $x=0,x=\left(5+\sqrt{2}\right),x=5-\sqrt{2}$ .

real zero is $0$ is a rational zero and $x=\left(5+\sqrt{2}\right),x=5-\sqrt{2}$ are the irrational zeros and there is no complex zeros.

Given information:

The given function is $f\left(x\right)=x^4-10x^3+23x^2$ and the the zeros are $x=0,x=\left(5+\sqrt{2}\right),x=5-\sqrt{2}$ .

Concept used:

Factor theorem: A polynomial function has a factor $x-k$ if only if $f\left(x\right)=0$

Rational number can be written in the format $\frac{p}{q}$ ,and An irrational number is a real number that cannot be expressed as a ratio of integers.

Calculations:

The function

  $f\left(x\right)=x^4-10x^3+23x^2$ by using ration function theorem can find the zeros and the zeros are $x=0,x=\left(5+\sqrt{2}\right),x=5-\sqrt{2}$ .

Here real zero is $0$ is a rational zero and $x=\left(5+\sqrt{2}\right),x=5-\sqrt{2}$ are the irrational zeros and there is no complex zeros.