Question

In Problems 39–56, find the real zeros of f. Use the real zeros to factor f.

$f\left(x\right)=2x^4-3x^3-21x^2-2x+24$

Short answer

The real zeros of fare $-2,-\frac{3}{2},1,and4.$

The factored form of f is$(2x+3)(x+2)(x-4)(x-1).$

Step-by-step solution

Step 1. Finding the possible number of zeros 

The given function is $f\left(x\right)=2x^4-3x^3-21x^2-2x+24$.

The given function is of degree four, so it has at most four real zeros.

Step 2. Use the rational zero theorem

As all the coefficients are integers so we use the rational zeros theorem.

The factors of the constant term $24$are

role="math" localid="1646129775343" $p:\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 8,\pm 12,\pm 24$

The factors of the leading coefficient $2$are

$q:\pm 1,\pm 2$

So, the possible rational zeros are

role="math" localid="1646129849786" $\frac{p}{q}:\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 8,\pm 12,\pm 24,\pm \frac{1}{2},\pm \frac{3}{2}$

Step 3. Graph the polynomial function  

The graph is

From the graph, we can depict that it has four roots.

Step 4. Finding the first factor 

Since $1$appears to be zero and a potential rational zero also.

By evaluating we get,$f\left(1\right)=0$

Thus, $(x-1)$is a factor of f.

Now, let's use synthetic division to factor f

So, factor fis$(x-1)(2x^3-x^2-22x-24).$

Step 5. Finding the second factor

It appears that $4$is a zero and a potential rational zero also.

By evaluating we get, $f\left(4\right)=0$

Thus, $(x-4)$ is a factor of f.

By using synthetic division we get,

So, factor fis$(x-1)(x-4)(2x^2+7x+6).$

Step 6. Factor the depressed polynomial

Put depressed equation to zero and factor by grouping

$$\begin{gathered} 2x^2+7x+6=0 \\ 2x^2+4x+3x+6=0 \\ 2x(x+2)+3(x+2)=0 \\ (2x+3)(x+2)=0 \end{gathered}$$

Therefore, the other zeros are

localid="1646132579858" $$\begin{gathered} 2x+3=0 \\ x=-\frac{3}{2} \\ andx+2=0 \\ x=-2 \end{gathered}$$

The factored form of f is$(2x+3)(x+2)(x-4)(x-1).$