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2. Factorize the following polynomials using the 'perfect square' form from the special product method.

(a) [tex]\(4x^4 + 8x^2 + 4\)[/tex]

Let [tex]\(a = 2x^2\)[/tex]:

[tex]\[
\begin{aligned}
4x^4 + 8x^2 + 4 &= (2x^2)^2 + 2 \cdot 2x^2 \cdot 2 + 2^2 \\
&= (2x^2 + 2)^2 \\
&= \left(2x^2 + 2\right)\left(2x^2 + 2\right)
\end{aligned}
\][/tex]

(b) [tex]\(x^4 + 6x^2 + 9\)[/tex]

Let [tex]\(a = x^2\)[/tex] and [tex]\(b = 3\)[/tex]:

[tex]\[
\begin{aligned}
x^4 + 6x^2 + 9 &= (x^2)^2 + 2 \cdot x^2 \cdot 3 + 3^2 \\
&= (x^2 + 3)^2
\end{aligned}
\][/tex]

(c) [tex]\(9x^4 - 12x^2 + 4\)[/tex]

Let [tex]\(a = 3x^2\)[/tex] and [tex]\(b = 2\)[/tex]:

[tex]\[
\begin{aligned}
9x^4 - 12x^2 + 4 &= (3x^2)^2 - 2 \cdot 3x^2 \cdot 2 + 2^2 \\
&= (3x^2 - 2)^2
\end{aligned}
\][/tex]

(d) [tex]\(25x^6 - 60x^3 + 36\)[/tex]

Let [tex]\(a = 5x^3\)[/tex] and [tex]\(b = 6\)[/tex]:

[tex]\[
\begin{aligned}
25x^6 - 60x^3 + 36 &= (5x^3)^2 - 2 \cdot 5x^3 \cdot 6 + 6^2 \\
&= (5x^3 - 6)^2
\end{aligned}
\][/tex]

Sagot :

GinnyAnsweridn
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.
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