Certainly! Let's use algebraic identities to find each of the given products step-by-step.
### (i) [tex]\((x+5)(x+2)\)[/tex]
To find the product of [tex]\((x+5)(x+2)\)[/tex], we use the distributive property (also known as the FOIL method in this case):
[tex]\[
(x+5)(x+2) = x \cdot x + x \cdot 2 + 5 \cdot x + 5 \cdot 2
\][/tex]
Simplifying each term:
[tex]\[
= x^2 + 2x + 5x + 10
\][/tex]
Combine the like terms ([tex]\(2x + 5x\)[/tex]):
[tex]\[
= x^2 + 7x + 10
\][/tex]
So, the result is:
[tex]\[
(x+5)(x+2) = x^2 + 7x + 10
\][/tex]
### (ii) [tex]\((x-5)(x-5)\)[/tex]
This product represents the square of a binomial. We use the identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
(x-5)^2 = x^2 - 2 \cdot x \cdot 5 + 5^2
\][/tex]
Simplifying the terms:
[tex]\[
= x^2 - 10x + 25
\][/tex]
So, the result is:
[tex]\[
(x-5)(x-5) = x^2 - 10x + 25
\][/tex]
### (iii) [tex]\((3x+2)(3x-2)\)[/tex]
This product represents the difference of squares. We use the identity [tex]\((a+b)(a-b) = a^2 - b^2\)[/tex]:
[tex]\[
(3x+2)(3x-2) = (3x)^2 - (2)^2
\][/tex]
Simplifying the terms:
[tex]\[
= 9x^2 - 4
\][/tex]
So, the result is:
[tex]\[
(3x+2)(3x-2) = 9x^2 - 4
\][/tex]
### (iv) [tex]\(\left(x^2 + \frac{1}{x^2}\right)\left(x^2 - \frac{1}{x^2}\right)\)[/tex]
This product also represents the difference of squares, where [tex]\(a = x^2\)[/tex] and [tex]\(b = \frac{1}{x^2}\)[/tex]:
[tex]\[
\left(x^2 + \frac{1}{x^2}\right)\left(x^2 - \frac{1}{x^2}\right) = (x^2)^2 - \left(\frac{1}{x^2}\right)^2
\][/tex]
Simplifying the terms:
[tex]\[
= x^4 - \frac{1}{x^4}
\][/tex]
So, the result is:
[tex]\[
\left(x^2 + \frac{1}{x^2}\right)\left(x^2 - \frac{1}{x^2}\right) = x^4 - \frac{1}{x^4}
\][/tex]
### (v) [tex]\((1+x)(1+x)\)[/tex]
This product represents the square of a binomial. We use the identity [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]:
[tex]\[
(1 + x)^2 = 1^2 + 2 \cdot 1 \cdot x + x^2
\][/tex]
Simplifying the terms:
[tex]\[
= 1 + 2x + x^2
\][/tex]
So, the result is:
[tex]\[
(1+x)(1+x) = 1 + 2x + x^2
\][/tex]
By following these steps and using the appropriate algebraic identities, we have computed the given products accurately.
