IDNLearn.com provides a user-friendly platform for finding and sharing accurate answers. Discover in-depth and trustworthy answers from our extensive network of knowledgeable professionals.
Sagot :
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].
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].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.