Chapter 4: Q. 67 (page 210)

In Problems 63–72, find the real solutions of each equation.
$3 x^{3} - x^{2} - 15 x + 5 = 0$

Short Answer

Expert verified

The real solution of the equation are $\frac{1}{3}, \sqrt{5}, - \sqrt{5} .$

Step by step solution

Step 1. Finding the possible number of zeros

The given equation is $3 x^{3} - x^{2} - 15 x + 5 = 0$

The given function is of degree three, so it has at most three 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 $5$ are

$p : \pm 1, \pm 5$

The factors of the leading coefficient $3$ are

$q : \pm 1, \pm 3$

So, the possible rational zeros are

role="math" localid="1646191718276" $\frac{p}{q} : \pm 1, \pm 5, \pm \frac{1}{3}, \pm \frac{5}{3}$

Step 3. Use the synthetic division

Let's test the potential zero $1,$

Since the remainder is not zero, it is not a factor.

Step 4. Finding the zero

Now, let's test the potential zero $\frac{1}{3}$ by using the synthetic division test

Since the remainder is zero, $(x - \frac{1}{3})$ is a factor.

The factor form is $(x - \frac{1}{3}) (3 x^{2} - 15) .$

Step 6. Use the zero product property

As the remaining zeros satisfy the depressed equation

So,

$3 x^{2} - 15 = 0 3 x^{2} = 15 x^{2} = 5 x = \pm \sqrt{5}$

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In Problems 63–72, find the real solutions of each equation.
$3 x^{3} - x^{2} - 15 x + 5 = 0$

Short Answer

Step by step solution

Step 1. Finding the possible number of zeros

Step 2. Use the rational zero theorem

Step 3. Use the synthetic division

Step 4. Finding the zero

Step 6. Use the zero product property

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In Problems 63–72, find the real solutions of each equation.3x3-x2-15x+5=0

Short Answer

Step by step solution

Step 1. Finding the possible number of zeros

Step 2. Use the rational zero theorem

Step 3. Use the synthetic division

Step 4. Finding the zero

Step 6. Use the zero product property

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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Theoretical and Mathematical Physics

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Decision Maths

Study anywhere. Anytime. Across all devices.

In Problems 63–72, find the real solutions of each equation.
$3 x^{3} - x^{2} - 15 x + 5 = 0$