# To verify: The algebraic expression a b = 1 2 [ ( a + b ) 2 − ( a 2 + b 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 129E

a.

To determine

## To verify: The algebraic expression ab=12[(a+b)2−(a2+b2)] .

Expert Solution

### Explanation of Solution

Given information:

The algebraic expression ab=12[(a+b)2(a2+b2)] .

Formula used:

The special factoring formula for perfect square which is mathematically expressed as,

(A+B)2=A2+2AB+B2

Proof:

Consider the function, ab=12[(a+b)2(a2+b2)] . The right hand side of the equation can be solved separately to verify the equation.

The right hand side of the equation is,

12[(a+b)2(a2+b2)]=12((a2+b2+2ab)(a2+b2))=12(a2+b2+2aba2b2)=12(2ab)=ab

Since, left hand side and right hand side are equal, therefore, the algebraic expression ab=12[(a+b)2(a2+b2)] is verified.

b.

To determine

Expert Solution

### Explanation of Solution

Given information:

The algebraic expression (a2+b2)2(a2b2)2=4a2b2 .

Formula used:

The special factoring formula for difference of squares, which is mathematically expressed as,

A2B2=(A+B)(AB)

Proof:

Consider the equation, (a2+b2)2(a2b2)2=4a2b2 . The left hand side of the equation can be solved separately to verify the equation.

The left hand side of equation is,

(a2+b2)2(a2b2)2=(a2+b2+a2b2)(a2+b2a2+b2)=(2a2)(2b2)=4a2b2

Since, left hand side and right hand side are equal, therefore, the algebraic expression (a2+b2)2(a2b2)2=4a2b2 is verified.

c.

To determine

Expert Solution

### Explanation of Solution

Given information:

Formula used:

The special factoring formula for perfect square which is mathematically expressed as,

(A+B)2=A2+2AB+B2

(AB)2=A22AB+B2

Proof:

Consider the equation, (a2+b2)(c2+d2)=(ac+bd)2+(adbc)2 . The right hand side of the equation can be solved separately to verify the equation.

The right hand side of the equation is,

Group the terms to take out common factors and simplify it further as,

Since, left hand side and right hand side are equal, therefore, the algebraic expression (a2+b2)(c2+d2)=(ac+bd)2+(adbc)2 is verified.

d.

To determine

### To calculate: The factor of the expression 4a2c2−(a2−b2+c2)2 .

Expert Solution

The factor of the expression 4a2c2(a2b2+c2)2 is a2(2c2a2+2b2)+b2(2c2b2)c4 .

### Explanation of Solution

Given information:

The expression 4a2c2(a2b2+c2)2 .

Formula used:

To factor out the common factor from a polynomial, find out the greatest common factor and express the polynomial as a product of the simpler ones.

Calculation:

Consider the given expression 4a2c2(a2b2+c2)2 .

Recall that to factor out the common factor from a polynomial, find out the greatest common factor and express the polynomial as a product of the simpler ones.

So, 4a2c2(a2b2+c2)2 can be written in simplified form as,

4a2c2(a2b2+c2)2=4a2c2(a2b2+c2)(a2b2+c2)=4a2c2a2(a2b2+c2)+b2(a2b2+c2)c2(a2b2+c2)=4a2c2a2a2+a2b2a2c2+b2a2b2b2+b2c2c2a2+b2c2c2c2=4a2c2a4+a2b2a2c2+b2a2b4+b2c2a2c2+b2c2c4

Simplify it further as,

4a2c2(a2b2+c2)2=4a2c2a4+a2b2a2c2+b2a2b4+b2c2a2c2+b2c2c4=4a2c2a4+2a2b22a2c2b4+2b2c2c4=(2a2c2a4+2a2b2)+(2b2c2b4)c4=a2(2c2a2+2b2)+b2(2c2b2)c4

Thus, the common factor of the expression 4a2c2(a2b2+c2)2 is a2(2c2a2+2b2)+b2(2c2b2)c4 .

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