Chapter 4: Q. 70 (page 210)

In Problems 63–72, find the real solutions of each equation.
$x^{4} - 2 x^{3} + 10 x^{2} - 18 x + 9 = 0$

Short Answer

Expert verified

The real solution of the equation is $1 .$

Step by step solution

Step 1. Finding the possible number of zeros

The given equation is $x^{4} - 2 x^{3} + 10 x^{2} - 18 x + 9 = 0$

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 $9$ are

$p : \pm 1, \pm 3, \pm 9$

The factors of the leading coefficient $1$ are

$q : \pm 1$

So, the possible rational zeros are

$\frac{p}{q} : \pm 1, \pm 3, \pm 9$

Step 3. Finding the zero

Let's test the potential zero $1$ by using substitution

Thus,

$f (1) = {(1)}^{4} - 2 {(1)}^{3} + 10 {(1)}^{2} - 18 (1) + 9 f (1) = 0$

Since the remainder is zero, $(x - 1)$ is a factor.

Step 4. Use synthetic division

Use synthetic division to factor equation,

Step 5. Use the zero product property

As the remaining zeros satisfy the depressed equation

So,

$x^{3} - x^{2} + 9 x - 9 = 0 x^{2} (x - 1) + 9 (x - 1) = 0 (x^{2} + 9) (x - 1) = 0$

Use the property

$x^{2} + 9 = 0 x^{2} = - 9$ and $x - 1 = 0 x = 1$

The square of a negative number is imaginary thus. it is not real.

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In Problems 63–72, find the real solutions of each equation.
$x^{4} - 2 x^{3} + 10 x^{2} - 18 x + 9 = 0$

Short Answer

Step by step solution

Step 1. Finding the possible number of zeros

Step 2. Use the rational zero theorem

Step 3. Finding the zero

Step 4. Use synthetic division

Step 5. Use the zero product property

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In Problems 63–72, find the real solutions of each equation.x4-2x3+10x2-18x+9=0

Short Answer

Step by step solution

Step 1. Finding the possible number of zeros

Step 2. Use the rational zero theorem

Step 3. Finding the zero

Step 4. Use synthetic division

Step 5. Use the zero product property

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Applied Mathematics

Statistics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 63–72, find the real solutions of each equation.
$x^{4} - 2 x^{3} + 10 x^{2} - 18 x + 9 = 0$