From science to arts, IDNLearn.com has the answers to all your questions. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.
Sagot :
To determine which polynomial is in standard form, we need to ensure that the terms are arranged in descending order of their degrees. Here's a step-by-step procedure for checking each polynomial:
1. Identify the degrees of the terms in each polynomial:
- Polynomial 1: [tex]\( 1 + 2x - 8x^2 + 6x^3 \)[/tex]
- Constant term: [tex]\(1\)[/tex] (degree [tex]\(0\)[/tex])
- Linear term: [tex]\(2x\)[/tex] (degree [tex]\(1\)[/tex])
- Quadratic term: [tex]\(-8x^2\)[/tex] (degree [tex]\(2\)[/tex])
- Cubic term: [tex]\(6x^3\)[/tex] (degree [tex]\(3\)[/tex])
- Polynomial 2: [tex]\( 2x^2 + 6x^3 - 9x + 12 \)[/tex]
- Constant term: [tex]\(12\)[/tex] (degree [tex]\(0\)[/tex])
- Linear term: [tex]\(-9x\)[/tex] (degree [tex]\(1\)[/tex])
- Quadratic term: [tex]\(2x^2\)[/tex] (degree [tex]\(2\)[/tex])
- Cubic term: [tex]\(6x^3\)[/tex] (degree [tex]\(3\)[/tex])
- Polynomial 3: [tex]\( 6x^3 + 5x - 3x^2 + 2 \)[/tex]
- Constant term: [tex]\(2\)[/tex] (degree [tex]\(0\)[/tex])
- Linear term: [tex]\(5x\)[/tex] (degree [tex]\(1\)[/tex])
- Quadratic term: [tex]\(-3x^2\)[/tex] (degree [tex]\(2\)[/tex])
- Cubic term: [tex]\(6x^3\)[/tex] (degree [tex]\(3\)[/tex])
- Polynomial 4: [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex]
- Constant term: [tex]\(5\)[/tex] (degree [tex]\(0\)[/tex])
- Linear term: [tex]\(-7x\)[/tex] (degree [tex]\(1\)[/tex])
- Quadratic term: [tex]\(4x^2\)[/tex] (degree [tex]\(2\)[/tex])
- Cubic term: [tex]\(2x^3\)[/tex] (degree [tex]\(3\)[/tex])
2. Arrange the terms in descending order of their degrees:
- Polynomial 1: [tex]\( 1 + 2x - 8x^2 + 6x^3 \)[/tex]
- Ordered: [tex]\( 6x^3 - 8x^2 + 2x + 1 \)[/tex]
- Polynomial 2: [tex]\( 2x^2 + 6x^3 - 9x + 12 \)[/tex]
- Ordered: [tex]\( 6x^3 + 2x^2 - 9x + 12 \)[/tex]
- Polynomial 3: [tex]\( 6x^3 + 5x - 3x^2 + 2 \)[/tex]
- Ordered: [tex]\( 6x^3 - 3x^2 + 5x + 2 \)[/tex]
- Polynomial 4: [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex]
- Ordered: [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex]
3. Check if the polynomials are already in the correct, standard form (descending order):
- Polynomial 1: [tex]\( 1 + 2x - 8x^2 + 6x^3 \)[/tex] is not in standard form.
- Polynomial 2: [tex]\( 2x^2 + 6x^3 - 9x + 12 \)[/tex] is not in standard form.
- Polynomial 3: [tex]\( 6x^3 + 5x - 3x^2 + 2 \)[/tex] is in standard form.
- Polynomial 4: [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex] is in standard form.
Thus, the polynomials that are in standard form are:
[tex]\[ 6x^3 + 5x - 3x^2 + 2 \][/tex]
[tex]\[ 2x^3 + 4x^2 - 7x + 5 \][/tex]
These polynomials have their terms arranged in descending order of their degrees.
1. Identify the degrees of the terms in each polynomial:
- Polynomial 1: [tex]\( 1 + 2x - 8x^2 + 6x^3 \)[/tex]
- Constant term: [tex]\(1\)[/tex] (degree [tex]\(0\)[/tex])
- Linear term: [tex]\(2x\)[/tex] (degree [tex]\(1\)[/tex])
- Quadratic term: [tex]\(-8x^2\)[/tex] (degree [tex]\(2\)[/tex])
- Cubic term: [tex]\(6x^3\)[/tex] (degree [tex]\(3\)[/tex])
- Polynomial 2: [tex]\( 2x^2 + 6x^3 - 9x + 12 \)[/tex]
- Constant term: [tex]\(12\)[/tex] (degree [tex]\(0\)[/tex])
- Linear term: [tex]\(-9x\)[/tex] (degree [tex]\(1\)[/tex])
- Quadratic term: [tex]\(2x^2\)[/tex] (degree [tex]\(2\)[/tex])
- Cubic term: [tex]\(6x^3\)[/tex] (degree [tex]\(3\)[/tex])
- Polynomial 3: [tex]\( 6x^3 + 5x - 3x^2 + 2 \)[/tex]
- Constant term: [tex]\(2\)[/tex] (degree [tex]\(0\)[/tex])
- Linear term: [tex]\(5x\)[/tex] (degree [tex]\(1\)[/tex])
- Quadratic term: [tex]\(-3x^2\)[/tex] (degree [tex]\(2\)[/tex])
- Cubic term: [tex]\(6x^3\)[/tex] (degree [tex]\(3\)[/tex])
- Polynomial 4: [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex]
- Constant term: [tex]\(5\)[/tex] (degree [tex]\(0\)[/tex])
- Linear term: [tex]\(-7x\)[/tex] (degree [tex]\(1\)[/tex])
- Quadratic term: [tex]\(4x^2\)[/tex] (degree [tex]\(2\)[/tex])
- Cubic term: [tex]\(2x^3\)[/tex] (degree [tex]\(3\)[/tex])
2. Arrange the terms in descending order of their degrees:
- Polynomial 1: [tex]\( 1 + 2x - 8x^2 + 6x^3 \)[/tex]
- Ordered: [tex]\( 6x^3 - 8x^2 + 2x + 1 \)[/tex]
- Polynomial 2: [tex]\( 2x^2 + 6x^3 - 9x + 12 \)[/tex]
- Ordered: [tex]\( 6x^3 + 2x^2 - 9x + 12 \)[/tex]
- Polynomial 3: [tex]\( 6x^3 + 5x - 3x^2 + 2 \)[/tex]
- Ordered: [tex]\( 6x^3 - 3x^2 + 5x + 2 \)[/tex]
- Polynomial 4: [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex]
- Ordered: [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex]
3. Check if the polynomials are already in the correct, standard form (descending order):
- Polynomial 1: [tex]\( 1 + 2x - 8x^2 + 6x^3 \)[/tex] is not in standard form.
- Polynomial 2: [tex]\( 2x^2 + 6x^3 - 9x + 12 \)[/tex] is not in standard form.
- Polynomial 3: [tex]\( 6x^3 + 5x - 3x^2 + 2 \)[/tex] is in standard form.
- Polynomial 4: [tex]\( 2x^3 + 4x^2 - 7x + 5 \)[/tex] is in standard form.
Thus, the polynomials that are in standard form are:
[tex]\[ 6x^3 + 5x - 3x^2 + 2 \][/tex]
[tex]\[ 2x^3 + 4x^2 - 7x + 5 \][/tex]
These polynomials have their terms arranged in descending order of their degrees.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. Discover insightful answers at IDNLearn.com. We appreciate your visit and look forward to assisting you again.
