# The factor of the expression x − 3 2 + 2 x − 1 2 + x 1 2 .

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.3, Problem 91E
To determine

## To calculate: The factor of the expression x−32+2x−12+x12 .

Expert Solution

The factor of the expression x32+2x12+x12 is 1x3(x+1)2 .

### Explanation of Solution

Given information:

The expression x32+2x12+x12 .

Formula used:

To factor an expression, split the terms in the expression into multiplication of simpler expressions, then take the common power out and group the expressions together.

To find the factor of the trinomial of the form x2+bx+c , find two numbers r and s such that sum of the numbers is equal to coefficient of x (r+s=b) and product of two numbers is equal to constant term (rs=c) , such that (x+r) and (x+s) are the factors of x2+bx+c .

x2+bx+c=x2+(r+s)x+rs=x2+rx+sx+rs=x(x+r)+s(x+r)=(x+r)(x+s)

Calculation:

Consider the given expression x32+2x12+x12 .

Recall that to factor an expression, split the terms in the expression into multiplication of simpler expressions, then take the common power out and group the expressions together.

The greatest common factor of these terms is x32 .

Apply it,

x32+2x12+x12=x32(1+2x12x32+x12x32)=x32(1+2x12(32)+x12(32))=1x3(1+2x+x2)=1x3(x2+2x+1)

Recall to find the factor of the trinomial of the form x2+bx+c , find two numbers r and s such that sum of the numbers is equal to coefficient of x (r+s=b) and product of two numbers is equal to constant term (rs=c) , such that (x+r) and (x+s) are the factors of x2+bx+c .

x2+bx+c=x2+(r+s)x+rs=x2+rx+sx+rs=x(x+r)+s(x+r)=(x+r)(x+s)

So, 1x3(x2+2x+1) will be further simplified as,

x32+2x12+x12=1x3(x2+2x+1)=1x3(x2+x+x+1)=1x3[x(x+1)+1(x+1)]=1x3(x+1)(x+1)

Simplify it further as,

x32+2x12+x12=1x3(x+1)(x+1)=1x3(x+1)2

Thus, the factor of expression x32+2x12+x12 is 1x3(x+1)2 .

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