Chapter 3.5, Problem 44PS

### Intermediate Algebra

10th Edition
Jerome E. Kaufmann + 1 other
ISBN: 9781285195728

Chapter
Section

### Intermediate Algebra

10th Edition
Jerome E. Kaufmann + 1 other
ISBN: 9781285195728
Textbook Problem

# For Problems 27 − 50 , factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don’t forget to look first for a common monomial factor. (Objective 1) 1 − 16 x 4 .

To determine

To factor:

The given polynomials completely and look first for a common monomial factor also indicate any that are not factorable using integers.

Explanation

Approach:

i) The Difference of Two Squares pattern states that a2âˆ’b2=(a+b)(aâˆ’b).

ii) Polynomial in the form of The Sum of Two Squares pattern like a2+b2 is not factorable using integers.

iii) Multiplication is commutative, so the order of writing the factors is not important. For example, (a+b)(aâˆ’b) can also be written as (aâˆ’b)(a+b).

The following steps have been followed to find out the complete factors:

Step-1: Convert the polynomial in the form of 2 squares with subtraction and compare with the difference of two squares pattern.

Step-2: Replace the value of a and b by the given binomial values in the difference of two squares pattern and simplify.

Step-3: If the simplified polynomial has the sum of two squares pattern and the difference of two squares pattern, then keep the sum of two squares pattern as it is. For the difference of two squares pattern, do the step-1 and step-2 again and write the complete answer.

Calculation:

1) The given polynomial is 1âˆ’16x4.

The above polynomial can be written as,

1âˆ’16x4=14âˆ’24x4=14âˆ’(2x)4=(12)2âˆ’((2x)2)2

Because, 14=12â‹…12=1â‹…1â‹…1â‹…1=1 and 24=22â‹…22=2â‹…2â‹…2â‹…2=16.

This equation has 2 squares and they are subtracted.

So, the above equation represents the difference of two squares pattern. That is, a2âˆ’b2=(a+b)(aâˆ’b).

2) Now, replace the value of the difference of two squares pattern by the given polynomial elements.

That is, a=12 and b=(2x)2

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