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Sagot :
To determine which polynomials are in standard form, we need to ensure that the terms of each polynomial are written in descending order of the exponents of the variable(s) involved.
Let's examine each polynomial individually:
1. [tex]\( x^2 + 3x + 2 \)[/tex]
- The terms are [tex]\( x^2 \)[/tex], [tex]\( 3x \)[/tex], and [tex]\( 2 \)[/tex]. The exponents [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex] are in descending order.
- This polynomial is in standard form.
2. [tex]\( q^3 - 15q + 12q^2 - 16 \)[/tex]
- The terms are [tex]\( q^3 \)[/tex], [tex]\( 12q^2 \)[/tex], [tex]\( -15q \)[/tex], and [tex]\( -16 \)[/tex]. The exponents [tex]\( 3 \)[/tex], [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex] are not in descending order (since [tex]\( 12q^2 \)[/tex] should come before [tex]\( -15q \)[/tex]).
- This polynomial is not in standard form.
3. [tex]\( 4a + a^2 + a - 2 \)[/tex]
- The terms are [tex]\( a^2 \)[/tex], [tex]\( 4a \)[/tex], [tex]\( a \)[/tex], and [tex]\( -2 \)[/tex]. The exponents [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex] are not in the proper position (since [tex]\( a^2 \)[/tex] should come before [tex]\( 4a \)[/tex]).
- This polynomial is not in standard form.
4. [tex]\( 3x^4 + 4x^3 - 3x^2 - 1 \)[/tex]
- The terms are [tex]\( 3x^4 \)[/tex], [tex]\( 4x^3 \)[/tex], [tex]\( -3x^2 \)[/tex], and [tex]\( -1 \)[/tex]. The exponents [tex]\( 4 \)[/tex], [tex]\( 3 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( 0 \)[/tex] are in descending order.
- This polynomial is in standard form.
5. [tex]\( 3t^3 + 3t^2 + 2t \)[/tex]
- The terms are [tex]\( 3t^3 \)[/tex], [tex]\( 3t^2 \)[/tex], and [tex]\( 2t \)[/tex]. The exponents [tex]\( 3 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( 1 \)[/tex] are in descending order.
- This polynomial is in standard form.
6. [tex]\( 14 + a^3 - 6a + 8a^2 \)[/tex]
- The terms are [tex]\( a^3 \)[/tex], [tex]\( 8a^2 \)[/tex], [tex]\( -6a \)[/tex], and [tex]\( 14 \)[/tex]. The exponents [tex]\( 3 \)[/tex], [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex] are not in descending order (since [tex]\( a^3 \)[/tex] should be the first term).
- This polynomial is not in standard form.
Therefore, the polynomials in standard form are:
[tex]\[ x^2 + 3x + 2 \][/tex]
[tex]\[ 3x^4 + 4x^3 - 3x^2 - 1 \][/tex]
[tex]\[ 3t^3 + 3t^2 + 2t \][/tex]
Let's examine each polynomial individually:
1. [tex]\( x^2 + 3x + 2 \)[/tex]
- The terms are [tex]\( x^2 \)[/tex], [tex]\( 3x \)[/tex], and [tex]\( 2 \)[/tex]. The exponents [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex] are in descending order.
- This polynomial is in standard form.
2. [tex]\( q^3 - 15q + 12q^2 - 16 \)[/tex]
- The terms are [tex]\( q^3 \)[/tex], [tex]\( 12q^2 \)[/tex], [tex]\( -15q \)[/tex], and [tex]\( -16 \)[/tex]. The exponents [tex]\( 3 \)[/tex], [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex] are not in descending order (since [tex]\( 12q^2 \)[/tex] should come before [tex]\( -15q \)[/tex]).
- This polynomial is not in standard form.
3. [tex]\( 4a + a^2 + a - 2 \)[/tex]
- The terms are [tex]\( a^2 \)[/tex], [tex]\( 4a \)[/tex], [tex]\( a \)[/tex], and [tex]\( -2 \)[/tex]. The exponents [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex] are not in the proper position (since [tex]\( a^2 \)[/tex] should come before [tex]\( 4a \)[/tex]).
- This polynomial is not in standard form.
4. [tex]\( 3x^4 + 4x^3 - 3x^2 - 1 \)[/tex]
- The terms are [tex]\( 3x^4 \)[/tex], [tex]\( 4x^3 \)[/tex], [tex]\( -3x^2 \)[/tex], and [tex]\( -1 \)[/tex]. The exponents [tex]\( 4 \)[/tex], [tex]\( 3 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( 0 \)[/tex] are in descending order.
- This polynomial is in standard form.
5. [tex]\( 3t^3 + 3t^2 + 2t \)[/tex]
- The terms are [tex]\( 3t^3 \)[/tex], [tex]\( 3t^2 \)[/tex], and [tex]\( 2t \)[/tex]. The exponents [tex]\( 3 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( 1 \)[/tex] are in descending order.
- This polynomial is in standard form.
6. [tex]\( 14 + a^3 - 6a + 8a^2 \)[/tex]
- The terms are [tex]\( a^3 \)[/tex], [tex]\( 8a^2 \)[/tex], [tex]\( -6a \)[/tex], and [tex]\( 14 \)[/tex]. The exponents [tex]\( 3 \)[/tex], [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex] are not in descending order (since [tex]\( a^3 \)[/tex] should be the first term).
- This polynomial is not in standard form.
Therefore, the polynomials in standard form are:
[tex]\[ x^2 + 3x + 2 \][/tex]
[tex]\[ 3x^4 + 4x^3 - 3x^2 - 1 \][/tex]
[tex]\[ 3t^3 + 3t^2 + 2t \][/tex]
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