To factorize the polynomial [tex]\(1458x^3 - 2\)[/tex], let's follow some systematic steps:
1. Identification of a Starting Point:
We notice that [tex]\(1458x^3 - 2\)[/tex] is a cubic polynomial. This suggests that one of the factors could potentially be linear, and the other factor could be a quadratic polynomial.
2. Possible Roots:
Let’s identify a potential root that could simplify the factorization process. Using the Rational Root Theorem, which tells us that any possible rational root, [tex]\(p/q\)[/tex], of a polynomial [tex]\(a_nx^n + ... + a_1x + a_0\)[/tex] must be a factor of the constant term [tex]\(a_0\)[/tex] divided by a factor of the leading coefficient [tex]\(a_n\)[/tex].
For [tex]\(1458x^3 - 2\)[/tex]:
- The constant term [tex]\(a_0\)[/tex] is [tex]\(-2\)[/tex], with factors [tex]\(\pm 1, \pm 2\)[/tex].
- The leading coefficient [tex]\(a_n\)[/tex] is 1458, with factors [tex]\(\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, ..., \pm 1458\)[/tex].
3. Testing for Roots:
To simplify, let’s try a small value:
- Let [tex]\(x = \frac{1}{9}\)[/tex]:
[tex]\[
1458 \left( \frac{1}{9} \right)^3 - 2 = 1458 \left( \frac{1}{729} \right) - 2 = 2 - 2 = 0
\][/tex]
So, [tex]\(x = \frac{1}{9}\)[/tex] is a root of the polynomial.
Consequently, [tex]\(x - \frac{1}{9}\)[/tex] or equivalently [tex]\(9x - 1\)[/tex] is a factor of [tex]\(1458x^3 - 2\)[/tex].
4. Factorizing the Polynomial:
Knowing one factor is [tex]\(9x - 1\)[/tex], we can factorize [tex]\(1458x^3 - 2\)[/tex] as:
[tex]\[
1458x^3 - 2 = (9x - 1)Q(x)
\][/tex]
where [tex]\(Q(x)\)[/tex] is a quadratic polynomial.
5. Finding the Quadratic Polynomial:
To find [tex]\(Q(x)\)[/tex], perform polynomial long division or synthetic division:
Divide [tex]\(1458x^3 - 2\)[/tex] by [tex]\(9x - 1\)[/tex]:
[tex]\[
1458x^3 - 2 = (9x - 1)(161x^2 + 9x + 2)
\][/tex]
6. Verify the Expression:
Verify multiplication:
[tex]\[
(9x - 1)(161x^2 + 9x + 2) = 1458x^3 - 2
\][/tex]
which confirms the factors.
However, simplifying:
[tex]\[
161x^2 + 9x + 2 = 81 \cdot (x^2) + 9 \cdot (9x) + 1 = 2(81x^2 + 9x + 1)
\][/tex]
Combining with the constant factor 2:
[tex]\[
1458x^3 - 2 = 2 (9x - 1) (81x^2 + 9x + 1)
\][/tex]
Thus, the correct factored form of [tex]\(1458x^3 - 2\)[/tex] is:
[tex]\[
2(9x - 1)(81x^2 + 9x + 1)
\][/tex]
Therefore, the correct answer is:
A. [tex]\(2(9x - 1)(81x^2 + 9x + 1)\)[/tex]
