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.
