Chapter 2.5, Problem 64E

To determine

To write : the polynomial as the product of linear factors and list all the zeros of function

Answer to Problem 64E

The function as product of linear factors: $\boxed{\boxed{g\left(x\right)=\left(x+5-2\sqrt{2}\right)\left(x+5+2\sqrt{2}\right)}}$

The zeros of the function are: $\boxed{\boxed{x=-5-2\sqrt{2};-5+2\sqrt{2}}}$

Explanation of Solution

Given information:

  $g\left(x\right)=x^2+10x+17$

Concept Involved:

Linear Factorization Theorem:If $f\left(x\right)$ is a polynomial of degree $n$ , where $n>0$ , then $f\left(x\right)$ has precisely $n$ linearfactors $f\left(x\right)=a_n\left(x-c_1\right)\left(x-c_2\right)...\left(x-c_n\right)$ where $c_1,c_2,...,c_n$ are complex numbers.

Complex Zeros Occur in Conjugate Pairs: Let f be a polynomial function that has real coefficients. If $a+bi$ , where $b\ne 0$ , is a zero of the function, then the complex conjugates $a-bi$ is alsoa zero of the function.

Calculation:

To find the zeros of the function we need to set the function to zero

  $g\left(x\right)=0$

  $x^2+10x+17=0$

We can solve this equation using completing the square method by subtracting17 on both sides of the equation

  $x^2+10x+17-17=0-17$

Simplify on both sides of the equation

  $x^2+10x=-17$

In order to make the left side expression as perfect square trinomial, we need to add square of half of coefficient of x on both sides

  $x^2+10x+{\left(10/2\right)}^2=-17+{\left(10/2\right)}^2$

Simplify on both sides of the equation of the equation

  $x^2+10x+25=8$

Write the left side as a perfect square

  ${\left(x+5\right)}^2=8$

Take square root on both sides

  $\sqrt{{\left(x+5\right)}^2}=\sqrt{8}$

Split the number in the right side of the equation in such a way that we can simplify further

  $\sqrt{{\left(x+5\right)}^2}=\sqrt{4}\cdot \sqrt{2}$

Simplifying square root on both sides of the equation

  $x+5=\pm 2\sqrt{2}$

Subtract 5 on both sides of the equation

  $x+5-5=-5\pm 2\sqrt{2}$

Simplify in left side of the equation

  $x=-5\pm 2\sqrt{2}$

List the zeros of the functions given:

  $x=-5+2\sqrt{2};x=-5-2\sqrt{2}$

If $x=k$ is zero, then $x-k$ will be the factor of the function.

So $x-\left(-5+2\sqrt{2}\right)$ & $x-\left(-5-2\sqrt{2}\right)$ are the factors of the function

Write the function $g\left(x\right)=x^2+10x+17$ as product of linear factors:

  $g\left(x\right)=\left(x+5-2\sqrt{2}\right)\left(x+5+2\sqrt{2}\right)$

Conclusion:

The zeros of the given function $g\left(x\right)=x^2+10x+17$ are: $x=-5-2\sqrt{2};-5+2\sqrt{2}$

The function written as product of linear factors: $g\left(x\right)=\left(x+5-2\sqrt{2}\right)\left(x+5+2\sqrt{2}\right)$