Question
7. Use the fact that the product of conjugates follows the pattern \( (a+b)(a-b)=a^{2}-b^{2} \) to quickly find th following products in standard form. \( \begin{array}{ll}\text { (g) }\left(x^{2}-2\right)\left(x^{2}+2\right) & \text { (i) }\left(x^{3}+4\right)\left(x^{3}-4\right)\end{array} \)
Ask by Colon Howell. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (x^2 - 2)(x^2 + 2) = x^4 - 4 \)
\( (x^3 + 4)(x^3 - 4) = x^6 - 16 \)
Solution
To solve the products using the pattern \( (a+b)(a-b) = a^2 - b^2 \), we will identify \( a \) and \( b \) in each case and apply the formula.
### Part (g): \( (x^2 - 2)(x^2 + 2) \)
Here, we can identify:
- \( a = x^2 \)
- \( b = 2 \)
Using the formula:
\[
(x^2 - 2)(x^2 + 2) = (x^2)^2 - (2)^2
\]
Now, we calculate:
\[
(x^2)^2 = x^4
\]
\[
(2)^2 = 4
\]
Putting it all together:
\[
(x^2 - 2)(x^2 + 2) = x^4 - 4
\]
### Part (i): \( (x^3 + 4)(x^3 - 4) \)
Here, we can identify:
- \( a = x^3 \)
- \( b = 4 \)
Using the formula:
\[
(x^3 + 4)(x^3 - 4) = (x^3)^2 - (4)^2
\]
Now, we calculate:
\[
(x^3)^2 = x^6
\]
\[
(4)^2 = 16
\]
Putting it all together:
\[
(x^3 + 4)(x^3 - 4) = x^6 - 16
\]
### Final Answers:
- (g) \( (x^2 - 2)(x^2 + 2) = x^4 - 4 \)
- (i) \( (x^3 + 4)(x^3 - 4) = x^6 - 16 \)
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Mind Expander
To tackle the first product \((x^{2}-2)(x^{2}+2)\), we can see that this follows the pattern \( (a+b)(a-b) = a^2 - b^2 \) where \(a = x^2\) and \(b = 2\). Applying this, we get: \[ (x^{2})^{2} - (2)^{2} = x^{4} - 4. \] So, the result of \((x^{2}-2)(x^{2}+2)\) in standard form is \(x^{4} - 4\). For the second product \((x^{3}+4)(x^{3}-4)\), we again match it to the same pattern with \(a = x^3\) and \(b = 4\). Thus, \[ (x^{3})^{2} - (4)^{2} = x^{6} - 16. \] Hence, in standard form, \((x^{3}+4)(x^{3}-4) = x^{6} - 16\).
