# The factors of the polynomial x 4 + x 2 − 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 137E

(a)

To determine

## To calculate: The factors of the polynomial x4+x2−2 .

Expert Solution

The polynomial x4+x22 is factorized as (x2+2)(x1)(x+1) .

### Explanation of Solution

Given information:

The polynomial x4+x22 .

Formula used:

To factor a polynomial split the middle term in such a way that sum of two numbers is the middle term and product of two numbers is same is product of first and last term.

The difference of square of two numbers a and b is a2b2=(a+b)(ab) .

Calculation:

Consider the polynomial x4+x22 .

Recall that to factor a polynomial split the middle term in such a way that sum of two numbers is the middle term and product of two numbers is same is product of first and last term.

Apply it,

x4+x22=x4+2x2x22=x2(x2+2)1(x2+2)=(x2+2)(x21)

Recall that the difference of square of two numbers a and b is a2b2=(a+b)(ab) .

Apply it,

x4+x22=x4+2x2x22=x2(x2+2)1(x2+2)=(x2+2)(x21)=(x2+2)(x1)(x+1)

Thus, the polynomial x4+x22 is factorized as (x2+2)(x1)(x+1) .

(b)

To determine

### To calculate: The factors of the polynomial x4+2x2+9 .

Expert Solution

The polynomial x4+2x2+9 is factorized as (x22x+3)(x2+2x+3) .

### Explanation of Solution

Given information:

The polynomial x4+2x2+9 .

Formula used:

To factor a polynomial add and subtract the terms to make it a perfect square.

The square of sum of two numbers a and b is (a+b)2=a2+b2+2ab .

The difference of square of two numbers a and b is a2b2=(a+b)(ab) .

Calculation:

Consider the polynomial x4+2x2+9 .

Recall that to factor a polynomial add and subtract the terms to make it a perfect square.

x4+2x2+9=x4+2x2+4x2+94x2=(x4+6x2+9)4x2

Now, factor the perfect square, square of sum of two numbers a and b is (a+b)2=a2+b2+2ab .

Simplify it further as,

x4+2x2+9=x4+2x2+4x2+94x2=(x4+6x2+9)4x2=(x2+3)24x2

Recall that the difference of square of two numbers a and b is a2b2=(a+b)(ab) .

Apply it,

x4+2x2+9=(x2+32x)(x2+3+2x)=(x22x+3)(x2+2x+3)

Thus, the polynomial x4+2x2+9 is factorized as (x22x+3)(x2+2x+3) .

(c)

To determine

### To calculate: The factors of the polynomial x4+4x2+16 .

Expert Solution

The polynomial x4+4x2+16 is factorized as (x22x+4)(x2+2x+4) .

### Explanation of Solution

Given information:

The polynomial x4+4x2+16 .

Formula used:

To factor a polynomial add and subtract the terms to make it a perfect square.

The square of sum of two numbers a and b is (a+b)2=a2+b2+2ab .

The difference of square of two numbers a and b is a2b2=(a+b)(ab) .

Calculation:

Consider the polynomial x4+4x2+16 .

Recall that to factor a polynomial add and subtract the terms to make it a perfect square.

x4+4x2+16=x4+4x2+4x2+164x2=(x4+8x2+42)4x2

Now, factor the perfect square, square of sum of two numbers a and b is (a+b)2=a2+b2+2ab .

Simplify it further as,

x4+4x2+16=x4+4x2+4x2+164x2=(x4+8x2+42)4x2=(x2+4)24x2

Recall that the difference of square of two numbers a and b is a2b2=(a+b)(ab) .

Apply it,

x4+4x2+16=(x2+42x)(x2+4+2x)=(x22x+4)(x2+2x+4)

Thus, the polynomial x4+4x2+16 is factorized as (x22x+4)(x2+2x+4) .

(d)

To determine

### To calculate: The factors of the polynomial x4+2x2+1 .

Expert Solution

The polynomial x4+2x2+1 is factorized as (x2+1)2 .

### Explanation of Solution

Given information:

The polynomial x4+2x2+1 .

Formula used:

To factor a polynomial add and subtract the terms to make it a perfect square.

The square of sum of two numbers a and b is (a+b)2=a2+b2+2ab .

The difference of square of two numbers a and b is a2b2=(a+b)(ab) .

Calculation:

Consider the polynomial x4+2x2+1 .

Rewrite the polynomial as, (x2)2+2(1)x2+12

Now, factor the perfect square, square of sum of two numbers a and b is (a+b)2=a2+b2+2ab .

(x2)2+2(1)x2+12=(x2+1)2

Thus, the polynomial x4+2x2+1 is factorized as (x2+1)2 .

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