Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To factor the polynomial [tex]\(12x^4 - 6x^5 + 18x^3\)[/tex] with the greatest common factor, follow these steps:
1. Identify the common factor: First, identify the greatest common factor (GCF) of all the terms in the polynomial.
- The terms are [tex]\(12x^4\)[/tex], [tex]\(-6x^5\)[/tex], and [tex]\(18x^3\)[/tex].
- The numerical coefficients are 12, -6, and 18. The GCF of these coefficients is 6.
- Each term also contains a factor of [tex]\(x\)[/tex]. The smallest power of [tex]\(x\)[/tex] in these terms is [tex]\(x^3\)[/tex]. Therefore, the GCF also includes [tex]\(x^3\)[/tex].
- Thus, the GCF of the entire polynomial is [tex]\(6x^3\)[/tex].
2. Factor out the GCF: Divide each term of the polynomial by the GCF [tex]\(6x^3\)[/tex].
- For [tex]\(12x^4\)[/tex]:
[tex]\[ \frac{12x^4}{6x^3} = 2x \][/tex]
- For [tex]\(-6x^5\)[/tex]:
[tex]\[ \frac{-6x^5}{6x^3} = -x^2 \][/tex]
- For [tex]\(18x^3\)[/tex]:
[tex]\[ \frac{18x^3}{6x^3} = 3 \][/tex]
3. Rewrite the polynomial: After factoring out [tex]\(6x^3\)[/tex], you get:
[tex]\[ 12x^4 - 6x^5 + 18x^3 = 6x^3(2x - x^2 + 3) \][/tex]
4. Reorganize the expression in a common form: Let's rewrite the polynomial in a standard form with exponents in descending order:
[tex]\[ -6x^5 + 12x^4 + 18x^3 = -6x^3(x^2 - 2x - 3) \][/tex]
5. Further factorization: Notice that you can factor the quadratic expression [tex]\(x^2 - 2x - 3\)[/tex] further:
- Solve the quadratic equation [tex]\(x^2 - 2x - 3 = 0\)[/tex].
- The roots of the equation are [tex]\(x = 3\)[/tex] and [tex]\(x = -1\)[/tex], so the quadratic can be factored as [tex]\((x - 3)(x + 1)\)[/tex].
6. Final Factorization: Substitute back these factors into the expression:
[tex]\[ -6x^5 + 12x^4 + 18x^3 = -6x^3(x - 3)(x + 1) \][/tex]
Thus, the completely factored form of the polynomial [tex]\(12x^4 - 6x^5 + 18x^3\)[/tex] is:
[tex]\[ -6x^5 + 12x^4 + 18x^3 = -6x^3(x - 3)(x + 1) \][/tex]
1. Identify the common factor: First, identify the greatest common factor (GCF) of all the terms in the polynomial.
- The terms are [tex]\(12x^4\)[/tex], [tex]\(-6x^5\)[/tex], and [tex]\(18x^3\)[/tex].
- The numerical coefficients are 12, -6, and 18. The GCF of these coefficients is 6.
- Each term also contains a factor of [tex]\(x\)[/tex]. The smallest power of [tex]\(x\)[/tex] in these terms is [tex]\(x^3\)[/tex]. Therefore, the GCF also includes [tex]\(x^3\)[/tex].
- Thus, the GCF of the entire polynomial is [tex]\(6x^3\)[/tex].
2. Factor out the GCF: Divide each term of the polynomial by the GCF [tex]\(6x^3\)[/tex].
- For [tex]\(12x^4\)[/tex]:
[tex]\[ \frac{12x^4}{6x^3} = 2x \][/tex]
- For [tex]\(-6x^5\)[/tex]:
[tex]\[ \frac{-6x^5}{6x^3} = -x^2 \][/tex]
- For [tex]\(18x^3\)[/tex]:
[tex]\[ \frac{18x^3}{6x^3} = 3 \][/tex]
3. Rewrite the polynomial: After factoring out [tex]\(6x^3\)[/tex], you get:
[tex]\[ 12x^4 - 6x^5 + 18x^3 = 6x^3(2x - x^2 + 3) \][/tex]
4. Reorganize the expression in a common form: Let's rewrite the polynomial in a standard form with exponents in descending order:
[tex]\[ -6x^5 + 12x^4 + 18x^3 = -6x^3(x^2 - 2x - 3) \][/tex]
5. Further factorization: Notice that you can factor the quadratic expression [tex]\(x^2 - 2x - 3\)[/tex] further:
- Solve the quadratic equation [tex]\(x^2 - 2x - 3 = 0\)[/tex].
- The roots of the equation are [tex]\(x = 3\)[/tex] and [tex]\(x = -1\)[/tex], so the quadratic can be factored as [tex]\((x - 3)(x + 1)\)[/tex].
6. Final Factorization: Substitute back these factors into the expression:
[tex]\[ -6x^5 + 12x^4 + 18x^3 = -6x^3(x - 3)(x + 1) \][/tex]
Thus, the completely factored form of the polynomial [tex]\(12x^4 - 6x^5 + 18x^3\)[/tex] is:
[tex]\[ -6x^5 + 12x^4 + 18x^3 = -6x^3(x - 3)(x + 1) \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.