Yes, your factorization is correct.

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In summary, the expression A^2 - B^2 + 16A + 64 can be factored using the grouping method into (A+8+B)(A+8-B). The expression is a sum of two expressions, A^2 + 16A + 64 and -B^2, and can also be written as the perfect square (A+8)^2.
  • #1
mathdad
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Precalculus by David Cohen, 3rd Edition
Chapter 1, Section 1.3.
Question 50.

Factor the expression.

A^2 - B^2 + 16A + 64

Factor by grouping method.

Group A = A^2 - B^2

Group A = (A - B)(A + B)

Group B = 16A + 64

Group B = 16(A + 4)

Group A + Group B

(A - B)(A + B) + 16(A + 4)

Correct?
 
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  • #2
RTCNTC said:
Precalculus by David Cohen, 3rd Edition
Chapter 1, Section 1.3.
Question 50.

Factor the expression.

A^2 - B^2 + 16A + 64

Factor by grouping method.

Group A = A^2 - B^2

Group A = (A - B)(A + B)

Group B = 16A + 64

Group B = 16(A + 4)

Group A + Group B

(A - B)(A + B) + 16(A + 4)

Correct?

No.
above is not factors. it is sum of 2 expressions

$A^2 - B^2 + 16A + 64 = A^2 + 16A + 64 - B^2 = (A+8)^2 - B^2 = (A+8+B)(A+8-B)$
 
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  • #3
kaliprasad said:
No.
above is not factors. it is sum of 2 expressions

$A^2 - B^2 + 16A + 64 = A^2 + 16A + 64 - B^2 = (A+8)^2 - B^2 = (A+8+B)(A+8-B)$

Why did you put B^2 to the far right?

Why did you put 16A + 64 in the center between A^2 and B^2?
 
  • #4
Because it was convenient. Kaliprasad recognized that [tex]A^2a+ 16A+ 64= (A+ 8)^2[/tex] is itself a "perfect square" so this could be written as a single "difference of squares"
 
  • #5
More factoring questions will be posted tomorrow from section 1.3 in David Cohen's precalculus textbook, third edition.
 

What is factorization?

Factorization is the process of finding the factors (numbers that can evenly divide) of a given number. In other words, it is the process of breaking down a number into smaller numbers that can be multiplied together to get the original number.

Why is it important to check if a factorization is correct?

It is important to check if a factorization is correct because it ensures the accuracy of your calculations. If the factorization is incorrect, the final result will also be incorrect.

How do I know if my factorization is correct?

You can check if your factorization is correct by multiplying the factors together and seeing if the result matches the original number. Additionally, you can use a calculator or a factorization tool to double-check your work.

What are some common mistakes when factorizing?

Some common mistakes when factorizing include missing factors, incorrect order of factors, and incorrect calculations. It is important to double-check your work and use a systematic approach to avoid these mistakes.

What are some tips for factorizing efficiently?

Some tips for factorizing efficiently include starting with the smallest possible factors, using divisibility rules, and breaking down larger numbers into smaller factors. It is also helpful to practice and familiarize yourself with common factors and factor patterns.

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