Chapter 4, Problem 5CRT

a.

To determine

List all possible rational zeros of $P$ .

Answer to Problem 5CRT

  $\boxed{2,4,-\frac{1}{2}}$

Explanation of Solution

Given information:

Let $p\left(x\right)=2x^3-11x^2+10x+8$

List all possible rational zeros of $P$ .

Calculation:

Consider the polynomial,

  $p\left(x\right)=2x^3-11x^2+10x+8$

Apply rational zero theorem to find all possible rational zeros of $P$ which will be in the form of,

  $\frac{p}{q}$

Where

  $p$ is the factor of constant coefficient.

  $q$ is the factor of leading coefficient.

Leading coefficient of given polynomial is $2$ we know that rational zeros will be,

Factor of $8$ /factor of $2$ .

  $\pm 1,\pm 2,\pm 4,\pm 8,\pm 0.5$

Now we can apply synthetic division to determine one of the zeros,

  $\begin{array}{l}2\underset{\_}{\overline{)2-11-10-8}}\\ \underset{\_}{4-14-8}\\ 2-7-4-0\end{array}$

The remainder is zero, conclude that $2$ is a zero of the function above.

By factorization,

  $\begin{array}{l}p\left(x\right)=2x^3-11x^2+10x+8\\ =\left(x-2\right)\left(2x^2-7x-4\right)\\ now,\\ \left(2x^2-7x-4\right)=\left(x-4\right)\left(2x+1\right)\\ \boxed{2,4,-\frac{1}{2}}\end{array}$

Hence, the zeros are $\boxed{2,4,-\frac{1}{2}}$

b.

To determine

Determine which of the numbers you listed in part (a) actually are zeros of $P$ .

Answer to Problem 5CRT

  $\boxed{p\left(x\right)=\left(x-2\right)\left(x-4\right)\left(2x+1\right)}$

Explanation of Solution

Given information:

Let $p\left(x\right)=2x^3-11x^2+10x+8$

Determine which of the numbers you listed in part (a) actually are zeros of $P$ .

Calculation:

Actual zeros of $P$ ,

  $\boxed{p\left(x\right)=\left(x-2\right)\left(x-4\right)\left(2x+1\right)}$

Hence, the zeros are $\boxed{p\left(x\right)=\left(x-2\right)\left(x-4\right)\left(2x+1\right)}$

c.

To determine

Factor $P$ completely.

Answer to Problem 5CRT

  $\boxed{2,4,-\frac{1}{2}}$

Explanation of Solution

Given information:

Let $p\left(x\right)=2x^3-11x^2+10x+8$

Factor $P$ completely.

Calculation:

By factorization,

  $\begin{array}{l}p\left(x\right)=2x^3-11x^2+10x+8\\ =\left(x-2\right)\left(2x^2-7x-4\right)\\ now,\\ \left(2x^2-7x-4\right)=\left(x-4\right)\left(2x+1\right)\\ \boxed{2,4,-\frac{1}{2}}\end{array}$

Hence, the solution is $\boxed{2,4,-\frac{1}{2}}$

d.

To determine

Sketch a graph of $P$ .

Answer to Problem 5CRT

  

Explanation of Solution

Given information:

Let $p\left(x\right)=2x^3-11x^2+10x+8$

Sketch a graph of $P$ .

Calculation:

The graph of $P$ is as shown,

  

Hence, the solution is graph.