Certainly! Let's look at each expression and the corresponding values they produce when [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]. We'll use these values to help us identify each expression's factored form.
1. Expression (5x - 7)(5x + 7):
- Factoring [tex]\( (5x - 7)(5x + 7) \)[/tex] gives us [tex]\( 25x^2 - 49 \)[/tex].
- Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[
(5 \cdot 1 - 7)(5 \cdot 1 + 7) = (5 - 7)(5 + 7) = (-2)(12) = -24
\][/tex]
- Result: -24
[tex]\[
- Therefore, the value of \( (5x - 7)(5x + 7) \) at \( x = 1 \) is -24.
2. Expression (x + 1)(x - 1):
- Factoring \( (x+1)(x-1) \) simplifies to \( x^2 - 1 \).
- Substitute \( x = 1 \):
\[
(1 + 1)(1 - 1) = 2 \cdot 0 = 0
\][/tex]
- Result: 0
[tex]\[
- Therefore, the value of \( (x+1)(x-1) \) at \( x = 1 \) is 0.
3. Expression (3x + 1)^2:
- Factoring \( (3x + 1)^2 \) gives us \( 9x^2 + 6x + 1 \).
- Substitute \( x = 1 \):
\[
(3 \cdot 1 + 1)^2 = (3 + 1)^2 = 4^2 = 16
\][/tex]
- Result: 16
[tex]\[
- Therefore, the value of \( (3x + 1)^2 \) at \( x = 1 \) is 16.
4. Expression (5x - 7)^2:
- Factoring \( (5x - 7)^2 \) gives us \( 25x^2 - 70x + 49 \).
- Substitute \( x = 1 \):
\[
(5 \cdot 1 - 7)^2 = (5 - 7)^2 = (-2)^2 = 4
\][/tex]
- Result: 4
[tex]\[
- Therefore, the value of \( (5x - 7)^2 \) at \( x = 1 \) is 4.
5. Expression (3x + 1)(3x - 1):
- Factoring \( (3x + 1)(3x - 1) \) gives us \( 9x^2 - 1 \).
- Substitute \( x = 1 \):
\[
(3 \cdot 1 + 1)(3 \cdot 1 - 1) = (3 + 1)(3 - 1) = 4 \cdot 2 = 8
\][/tex]
- Result: 8
[tex]\[
- Therefore, the value of \( (3x + 1)(3x - 1) \) at \( x = 1 \) is 8.
6. Expression (x - 1)^2:
- Factoring \( (x - 1)^2 \) gives us \( x^2 - 2x + 1 \).
- Substitute \( x = 2 \):
\[
(2 - 1)^2 = 1^2 = 1
\][/tex]
- Result: 1
\[
- Therefore, the value of [tex]\( (x - 1)^2 \)[/tex] at [tex]\( x = 2 \)[/tex] is 1.
The values we calculated:
- [tex]\((-24, 0, 16, 4, 8, 1)\)[/tex]
These values correspond to the expressions:
a. [tex]\((5x - 7)(5x + 7)\)[/tex] produces [tex]\(-24\)[/tex]
b. [tex]\((x + 1)(x - 1)\)[/tex] produces [tex]\(0\)[/tex]
c. [tex]\((3x + 1)^2\)[/tex] produces [tex]\(16\)[/tex]
d. [tex]\((5x - 7)^2\)[/tex] produces [tex]\(4\)[/tex]
e. [tex]\((3x + 1)(3x - 1)\)[/tex] produces [tex]\(8\)[/tex]
f. [tex]\((x - 1)^2\)[/tex] produces [tex]\(1\)[/tex]
So, each of the given expressions matches up as follows:
- [tex]\( (5 x - 7)(5 x + 7) \)[/tex] matches to a. [tex]\(-24\)[/tex]
- [tex]\( (x + 1)(x - 1) \)[/tex] matches to b. [tex]\(0\)[/tex]
- [tex]\( (3 x + 1)^2 \)[/tex] matches to c. [tex]\(16\)[/tex]
- [tex]\( (5 x - 7)^2 \)[/tex] matches to d. [tex]\(4\)[/tex]
- [tex]\( (3 x + 1)(3 x - 1) \)[/tex] matches to e. [tex]\(8\)[/tex]
- [tex]\( (x - 1)^2 \)[/tex] matches to f. [tex]\(1\)[/tex]
