Let's factorize the polynomial [tex]\(15 x^2 - x - 2\)[/tex]. We aim to express it in the form of [tex]\((ax + b)(cx + d)\)[/tex].
1. Start by identifying the polynomial:
[tex]\[
15 x^2 - x - 2
\][/tex]
2. We seek two binomials [tex]\((ax + b)(cx + d)\)[/tex] that, when expanded, will yield the original polynomial. Let's assume:
[tex]\[
(3x + 1)(5x - 2)
\][/tex]
3. Expand the binomials to verify:
Step-by-step expansion:
[tex]\[
(3x + 1)(5x - 2)
\][/tex]
Multiply [tex]\(3x\)[/tex] by [tex]\(5x\)[/tex]:
[tex]\[
3x \cdot 5x = 15x^2
\][/tex]
Multiply [tex]\(3x\)[/tex] by [tex]\(-2\)[/tex]:
[tex]\[
3x \cdot (-2) = -6x
\][/tex]
Multiply [tex]\(1\)[/tex] by [tex]\(5x\)[/tex]:
[tex]\[
1 \cdot 5x = 5x
\][/tex]
Multiply [tex]\(1\)[/tex] by [tex]\(-2\)[/tex]:
[tex]\[
1 \cdot (-2) = -2
\][/tex]
4. Combine the results:
[tex]\[
15x^2 + (-6x) + 5x - 2 = 15x^2 - x - 2
\][/tex]
The expanded form matches the original polynomial.
Therefore, the factorized form of [tex]\(15 x^2 - x - 2\)[/tex] is:
[tex]\[
(3x + 1)(5x - 2)
\][/tex]
