To factorize the polynomial [tex]\( 4x^2 - 36x + 81 \)[/tex] into the form [tex]\( (a x - b)^2 \)[/tex], let's break down the steps:
1. Identify the Standard Form of the Quadratic Polynomial:
The polynomial we have is [tex]\( 4x^2 - 36x + 81 \)[/tex].
2. Form Hypothesis of Factored Form:
We are told to factor this into the form [tex]\( (a x - b)^2 \)[/tex].
3. Assume a Quadratic Factored Form:
Let's assume [tex]\( (a x - b)^2 \)[/tex].
4. Expand the Hypothetical Factored Form:
Expanding [tex]\( (a x - b)^2 \)[/tex] gives us:
[tex]\[
(a x - b)^2 = a^2 x^2 - 2abx + b^2
\][/tex]
5. Equate the Expanded Form with Given Polynomial:
Compare this with [tex]\( 4x^2 - 36x + 81 \)[/tex]:
- The coefficient of [tex]\( x^2 \)[/tex] matches: [tex]\( a^2 = 4 \)[/tex]
- The coefficient of [tex]\( x \)[/tex] matches: [tex]\( -2ab = -36 \)[/tex]
- The constant term matches: [tex]\( b^2 = 81 \)[/tex]
6. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
From [tex]\( a^2 = 4 \)[/tex]:
[tex]\[
a = 2 \quad \text{or} \quad a = -2
\][/tex]
From [tex]\( b^2 = 81 \)[/tex]:
[tex]\[
b = 9 \quad \text{or} \quad b = -9
\][/tex]
Using [tex]\( a = 2 \)[/tex], solve [tex]\( -2ab = -36 \)[/tex]:
[tex]\[
-2 \cdot 2 \cdot b = -36 \implies -4b = -36 \implies b = 9
\][/tex]
Using [tex]\( a = -2 \)[/tex], solve [tex]\( -2ab = -36 \)[/tex]:
[tex]\[
-2 \cdot (-2) \cdot b = -36 \implies 4b = -36 \implies b = -9
\][/tex]
Both pairs [tex]\( (a, b) = (2, 9) \)[/tex] and [tex]\( (a, b) = (-2, -9) \)[/tex] work for the polynomial.
7. Construct the Factored Form:
The factored form is:
[tex]\[
(2 x - 9)^2
\][/tex]
Therefore, the factored form of [tex]\( 4 x^2 - 36 x + 81 \)[/tex] is:
[tex]\[
(2 x - 9)^2
\][/tex]
Thus, when asked to write the polynomial in the form [tex]\( ([a] x - b)^2 \)[/tex]:
[tex]\[
\boxed{(2 x - 9)^2}
\][/tex]
