Craig Mills
10/22/2023 · Primary School
3. \( 9 e^{2}-4 \) 4. \( a^{4}-16 \) \( (a+b)^{2}-c^{2} \)
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Step-by-step Solution
Let's simplify and factor each of the given expressions one by one.
### 3. \( 9 e^{2}-4 \)
This expression is a difference of squares, which can be factored using the identity \( a^2 - b^2 = (a - b)(a + b) \).
Here, we can rewrite \( 9 e^{2} \) as \( (3e)^2 \) and \( 4 \) as \( 2^2 \):
\[
9 e^{2} - 4 = (3e)^2 - 2^2 = (3e - 2)(3e + 2)
\]
### 4. \( a^{4}-16 \)
This expression is also a difference of squares. We can rewrite \( a^{4} \) as \( (a^2)^2 \) and \( 16 \) as \( 4^2 \):
\[
a^{4} - 16 = (a^2)^2 - 4^2 = (a^2 - 4)(a^2 + 4)
\]
Now, we can further factor \( a^2 - 4 \) since it is also a difference of squares:
\[
a^2 - 4 = (a - 2)(a + 2)
\]
Thus, the complete factorization of \( a^{4} - 16 \) is:
\[
a^{4} - 16 = (a - 2)(a + 2)(a^2 + 4)
\]
### \( (a+b)^{2}-c^{2} \)
This expression is again a difference of squares. We can rewrite it as follows:
\[
(a + b)^{2} - c^{2} = ((a + b) - c)((a + b) + c)
\]
So, the factored form is:
\[
(a + b - c)(a + b + c)
\]
### Summary of Factored Forms
1. \( 9 e^{2} - 4 = (3e - 2)(3e + 2) \)
2. \( a^{4} - 16 = (a - 2)(a + 2)(a^2 + 4) \)
3. \( (a + b)^{2} - c^{2} = (a + b - c)(a + b + c) \)
Quick Answer
1. \( 9 e^{2} - 4 = (3e - 2)(3e + 2) \)
2. \( a^{4} - 16 = (a - 2)(a + 2)(a^2 + 4) \)
3. \( (a + b)^{2} - c^{2} = (a + b - c)(a + b + c) \)
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