Sure! Let's go through the steps to factorize the given polynomials using the concept of perfect squares.
### (a) [tex]\( x^4 + 6x^2 + 9 \)[/tex]
1. Identify the form [tex]\( a^2 + 2ab + b^2 = (a + b)^2 \)[/tex].
2. Let's set [tex]\( a = x^2 \)[/tex] and [tex]\( b = 3 \)[/tex].
3. Substitute these values to rewrite the polynomial:
[tex]\[
(x^2)^2 + 2 \cdot x^2 \cdot 3 + 3^2 = (x^2 + 3)^2
\][/tex]
Therefore, the factorized form is:
[tex]\[
\boxed{(x^2 + 3)^2}
\][/tex]
### (b) [tex]\( 9x^4 - 12x^2 + 4 \)[/tex]
1. Identify the form [tex]\( a^2 - 2ab + b^2 = (a - b)^2 \)[/tex].
2. Let's set [tex]\( a = 3x^2 \)[/tex] and [tex]\( b = 2 \)[/tex].
3. Substitute these values to rewrite the polynomial:
[tex]\[
(3x^2)^2 - 2 \cdot 3x^2 \cdot 2 + 2^2 = (3x^2 - 2)^2
\][/tex]
Therefore, the factorized form is:
[tex]\[
\boxed{(3x^2 - 2)^2}
\][/tex]
### (c) [tex]\( 25x^6 - 60x^3 + 36 \)[/tex]
1. Identify the form [tex]\( a^2 - 2ab + b^2 = (a - b)^2 \)[/tex] (Note that the middle term needs simplification).
2. Let's set [tex]\( a = 5x^3 \)[/tex] and [tex]\( b = 3 \)[/tex].
3. Substitute these values to rewrite the polynomial:
[tex]\[
(5x^3)^2 - 2 \cdot 5x^3 \cdot 3 + 3^2 = (5x^3 - 3)^2
\][/tex]
Therefore, the factorized form verifies to be:
[tex]\[
\boxed{(5x^3 - 3)^2}
\][/tex]
Each polynomial is transformed into a perfect square, which completes their factorization.
