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Chapter 4: Q. 70 (page 210)

In Problems 63鈥�72, find the real solutions of each equation.

x4-2x3+10x2-18x+9=0

Short Answer

Expert verified

The real solution of the equation is1.

Step by step solution

01

Step 1. Finding the possible number of zeros 

The given equation isx4-2x3+10x2-18x+9=0

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

02

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 9are

p:卤1,卤3,卤9

The factors of the leading coefficient 1are

q:卤1

So, the possible rational zeros are

pq:卤1,卤3,卤9

03

Step 3. Finding the zero 

Let's test the potential zero 1by using substitution

Thus,

f(1)=14-213+1012-181+9f(1)=0

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

04

Step 4. Use synthetic division 

Use synthetic division to factor equation,

05

Step 5. Use the zero product property 

As the remaining zeros satisfy the depressed equation

So,

x3鈭�x2+9x鈭�9=0x2x-1+9x-1=0x2+9x-1=0

Use the property

x2+9=0x2=-9and x-1=0x=1

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

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