Sure, let's factorize the given expressions one by one.
### (i) [tex]\( x^2 + 6xy + 9y^2 \)[/tex]
1. Look at the expression [tex]\(x^2 + 6xy + 9y^2\)[/tex].
2. Notice that it follows the pattern of a perfect square trinomial, which generally takes the form [tex]\(a^2 + 2ab + b^2 = (a + b)^2\)[/tex].
3. Here, we can identify [tex]\(a\)[/tex] as [tex]\(x\)[/tex] and [tex]\(b\)[/tex] as [tex]\(3y\)[/tex].
4. Write the expression in the form of [tex]\((x + 3y)^2\)[/tex].
So, the factorized form is:
[tex]\[ x^2 + 6xy + 9y^2 = (x + 3y)^2 \][/tex]
### (iv) [tex]\( 9x^2 - 24xy + 16y^2 \)[/tex]
1. Look at the expression [tex]\(9x^2 - 24xy + 16y^2\)[/tex].
2. This expression also follows the pattern of a perfect square trinomial, [tex]\(a^2 - 2ab + b^2 = (a - b)^2\)[/tex].
3. Identify [tex]\(a\)[/tex] as [tex]\(3x\)[/tex] and [tex]\(b\)[/tex] as [tex]\(4y\)[/tex].
4. Write the expression in the form of [tex]\((3x - 4y)^2\)[/tex].
So, the factorized form is:
[tex]\[ 9x^2 - 24xy + 16y^2 = (3x - 4y)^2 \][/tex]
### (vii) [tex]\( 1 - 2x + x^2 \)[/tex]
1. Look at the expression [tex]\(1 - 2x + x^2\)[/tex].
2. Notice that it follows the pattern of a perfect square trinomial, [tex]\(a^2 - 2ab + b^2 = (a - b)^2\)[/tex].
3. Identify [tex]\(a\)[/tex] as [tex]\(x\)[/tex] and [tex]\(b\)[/tex] as [tex]\(1\)[/tex]. Note that the terms can be reorganized as [tex]\(x^2 - 2x + 1\)[/tex].
4. Write the expression in the form of [tex]\((x - 1)^2\)[/tex].
So, the factorized form is:
[tex]\[ 1 - 2x + x^2 = (x - 1)^2 \][/tex]
In summary, the factorized forms of the given expressions are:
1. [tex]\( x^2 + 6xy + 9y^2 = (x + 3y)^2 \)[/tex]
2. [tex]\( 9x^2 - 24xy + 16y^2 = (3x - 4y)^2 \)[/tex]
3. [tex]\( 1 - 2x + x^2 = (x - 1)^2 \)[/tex]
