BuyFind

Precalculus: Mathematics for Calcu...

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

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

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

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

To verify: The algebraic expression (a2+b2)(c2+d2)=(ac+bd)2+(adbc)2 .

Expert Solution

Explanation of Solution

Given information:

The algebraic expression, (a2+b2)(c2+d2)=(ac+bd)2+(adbc)2 .

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,

  (ac+bd)2+(adbc)2=(ac)2+2(ac)(bd)+(bd)2+(ad)22(ad)(bc)+(bc)2=a2c2+2acbd+b2d2+a2d22adbc+b2c2=a2c2+b2d2+a2d2+b2c2

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

  (ac+bd)2+(adbc)2=a2c2+b2d2+a2d2+b2c2=(a2c2+a2d2)+(b2d2+b2c2)=a2(c2+d2)+b2(d2+c2)=(a2+b2)(c2+d2)

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(a2b2+c2)2 .

Expert Solution

Answer to Problem 129E

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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