To find the greatest common divisor (GCD) of the polynomials [tex]\(6x^2 - 13x + 6\)[/tex] and [tex]\(4x^2 - 4x - 3\)[/tex], we need to follow these steps:
1. Define the Polynomials:
Let's denote the first polynomial as [tex]\(f(x)\)[/tex] and the second polynomial as [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 6x^2 - 13x + 6
\][/tex]
[tex]\[
g(x) = 4x^2 - 4x - 3
\][/tex]
2. Factorize Both Polynomials:
We need to find the factors of each polynomial separately.
- For [tex]\(f(x) = 6x^2 - 13x + 6\)[/tex]:
We look for two numbers that multiply to [tex]\(6 \times 6 = 36\)[/tex] and add up to [tex]\(-13\)[/tex]. These numbers are [tex]\(-4\)[/tex] and [tex]\(-9\)[/tex]. So, we write:
[tex]\[
6x^2 - 13x + 6 = 6x^2 - 9x - 4x + 6
\][/tex]
Grouping the terms:
[tex]\[
= 3x(2x - 3) - 2(2x - 3) = (3x - 2)(2x - 3)
\][/tex]
- For [tex]\(g(x) = 4x^2 - 4x - 3\)[/tex]:
We look for two numbers that multiply to [tex]\(4 \times -3 = -12\)[/tex] and add up to [tex]\(-4\)[/tex]. These numbers are [tex]\(-6\)[/tex] and [tex]\(2\)[/tex]. So, we write:
[tex]\[
4x^2 - 4x - 3 = 4x^2 + 2x - 6x - 3
\][/tex]
Grouping the terms:
[tex]\[
= 2x(2x + 1) - 3(2x + 1) = (2x + 1)(2x - 3)
\][/tex]
3. Identify Common Factors:
Looking at both factorizations:
[tex]\[
f(x) = (3x - 2)(2x - 3)
\][/tex]
[tex]\[
g(x) = (2x + 1)(2x - 3)
\][/tex]
The common factor in both polynomials is [tex]\(2x - 3\)[/tex].
4. Conclusion:
Thus, the greatest common divisor (GCD) of the two polynomials is:
[tex]\[
2x - 3
\][/tex]
Therefore, the GCD of the polynomials [tex]\(6x^2 - 13x + 6\)[/tex] and [tex]\(4x^2 - 4x - 3\)[/tex] is [tex]\(2x - 3\)[/tex].
