Join IDNLearn.com and start exploring the answers to your most pressing questions. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.
Sagot :
To determine the completely factored form of [tex]\(d^4 - 81\)[/tex], we begin by recognizing that [tex]\(81\)[/tex] is a perfect square, as [tex]\(81 = 9^2\)[/tex]. Thus, we can rewrite the expression in a form that highlights this relationship:
[tex]\[ d^4 - 81 = d^4 - 9^2. \][/tex]
We notice that this fits the pattern of a difference of squares, [tex]\(a^2 - b^2\)[/tex], which factors into [tex]\((a - b)(a + b)\)[/tex]. Here, [tex]\(a = d^2\)[/tex] and [tex]\(b = 9\)[/tex]:
[tex]\[ d^4 - 81 = (d^2)^2 - 9^2 = (d^2 - 9)(d^2 + 9). \][/tex]
Next, we can factor the term [tex]\(d^2 - 9\)[/tex] further, as it is also a difference of squares. We apply the same pattern again, where [tex]\(d^2 - 9\)[/tex] can be written as:
[tex]\[ d^2 - 9 = (d - 3)(d + 3). \][/tex]
Thus, substituting back, we have:
[tex]\[ d^4 - 81 = (d^2 - 9)(d^2 + 9) = (d - 3)(d + 3)(d^2 + 9). \][/tex]
The term [tex]\(d^2 + 9\)[/tex] cannot be factored further as a real number expression since it does not fit the pattern for a difference of squares or any other recognizable factoring pattern for real numbers.
Hence, the completely factored form of [tex]\(d^4 - 81\)[/tex] is:
[tex]\[ (d - 3)(d + 3)(d^2 + 9). \][/tex]
[tex]\[ d^4 - 81 = d^4 - 9^2. \][/tex]
We notice that this fits the pattern of a difference of squares, [tex]\(a^2 - b^2\)[/tex], which factors into [tex]\((a - b)(a + b)\)[/tex]. Here, [tex]\(a = d^2\)[/tex] and [tex]\(b = 9\)[/tex]:
[tex]\[ d^4 - 81 = (d^2)^2 - 9^2 = (d^2 - 9)(d^2 + 9). \][/tex]
Next, we can factor the term [tex]\(d^2 - 9\)[/tex] further, as it is also a difference of squares. We apply the same pattern again, where [tex]\(d^2 - 9\)[/tex] can be written as:
[tex]\[ d^2 - 9 = (d - 3)(d + 3). \][/tex]
Thus, substituting back, we have:
[tex]\[ d^4 - 81 = (d^2 - 9)(d^2 + 9) = (d - 3)(d + 3)(d^2 + 9). \][/tex]
The term [tex]\(d^2 + 9\)[/tex] cannot be factored further as a real number expression since it does not fit the pattern for a difference of squares or any other recognizable factoring pattern for real numbers.
Hence, the completely factored form of [tex]\(d^4 - 81\)[/tex] is:
[tex]\[ (d - 3)(d + 3)(d^2 + 9). \][/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.