Solved on Jan 12, 2024

Find the polynomials in standard form. Choose all correct options. $5x^5 + 4x^4 + 12x^3 - 23x + 15$ , $2x^4 - 12x^2 + 5x - 9$ , $5 - 3x$ , $4x^2 + 12x - 13x^3$ .

STEP 1

Assumptions
1. We are looking for polynomials in Standard Form.
2. Standard Form for a polynomial means the terms are written in descending order of their degree (the highest power of x comes first, followed by lower powers).
3. The coefficients of the terms can be any real number.

STEP 2

Examine the first polynomial to determine if it is in Standard Form.
$5 x^{5}+4 x^{4}+12 x^{3}-23 x+15$

STEP 3

Check if the terms of the first polynomial are in descending order by degree.
The terms are in order: $x^{5}$ , $x^{4}$ , $x^{3}$ , $x^{1}$ , and the constant term.

STEP 4

Confirm that the first polynomial is in Standard Form.
Since the terms are in descending order by degree, the first polynomial is in Standard Form.

STEP 5

Examine the second polynomial to determine if it is in Standard Form.
$2 x^{4}-12 x^{2}+5 x-9$

STEP 6

Check if the terms of the second polynomial are in descending order by degree.
The terms are in order: $x^{4}$ , $x^{2}$ , $x^{1}$ , and the constant term.

STEP 7

Confirm that the second polynomial is in Standard Form.
Since the terms are in descending order by degree, the second polynomial is in Standard Form.

STEP 8

Examine the third polynomial to determine if it is in Standard Form.
$5-3 x$

STEP 9

Check if the terms of the third polynomial are in descending order by degree.
The terms are in order: the constant term, and then $x^{1}$ .

STEP 10

Confirm that the third polynomial is not in Standard Form.
Since the terms are not in descending order by degree (the constant term should come after the $x$ term), the third polynomial is not in Standard Form.

STEP 11

Examine the fourth polynomial to determine if it is in Standard Form.
$4 x^{2}+12 x-13 x^{3}$

STEP 12

Check if the terms of the fourth polynomial are in descending order by degree.
The terms are in order: $x^{2}$ , $x^{1}$ , and then $x^{3}$ .

STEP 13

Confirm that the fourth polynomial is not in Standard Form.
Since the terms are not in descending order by degree (the $x^{3}$ term should come before the $x^{2}$ and $x$ terms), the fourth polynomial is not in Standard Form.

STEP 14

List all polynomials that are in Standard Form.
The polynomials in Standard Form are: $5 x^{5}+4 x^{4}+12 x^{3}-23 x+15$ $2 x^{4}-12 x^{2}+5 x-9$

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Find the polynomials in standard form. Choose all correct options. $5x^5 + 4x^4 + 12x^3 - 23x + 15$ , $2x^4 - 12x^2 + 5x - 9$ , $5 - 3x$ , $4x^2 + 12x - 13x^3$ .

STEP 1

Assumptions
1. We are looking for polynomials in Standard Form.
2. Standard Form for a polynomial means the terms are written in descending order of their degree (the highest power of x comes first, followed by lower powers).
3. The coefficients of the terms can be any real number.

STEP 2

Examine the first polynomial to determine if it is in Standard Form.
$5 x^{5}+4 x^{4}+12 x^{3}-23 x+15$

STEP 3

Check if the terms of the first polynomial are in descending order by degree.
The terms are in order: $x^{5}$ , $x^{4}$ , $x^{3}$ , $x^{1}$ , and the constant term.

STEP 4

Confirm that the first polynomial is in Standard Form.
Since the terms are in descending order by degree, the first polynomial is in Standard Form.

STEP 5

Examine the second polynomial to determine if it is in Standard Form.
$2 x^{4}-12 x^{2}+5 x-9$

STEP 6

Check if the terms of the second polynomial are in descending order by degree.
The terms are in order: $x^{4}$ , $x^{2}$ , $x^{1}$ , and the constant term.

STEP 7

Confirm that the second polynomial is in Standard Form.
Since the terms are in descending order by degree, the second polynomial is in Standard Form.

STEP 8

Examine the third polynomial to determine if it is in Standard Form.
$5-3 x$

STEP 9

Check if the terms of the third polynomial are in descending order by degree.
The terms are in order: the constant term, and then $x^{1}$ .

STEP 10

Confirm that the third polynomial is not in Standard Form.
Since the terms are not in descending order by degree (the constant term should come after the $x$ term), the third polynomial is not in Standard Form.

STEP 11

Examine the fourth polynomial to determine if it is in Standard Form.
$4 x^{2}+12 x-13 x^{3}$

STEP 12

Check if the terms of the fourth polynomial are in descending order by degree.
The terms are in order: $x^{2}$ , $x^{1}$ , and then $x^{3}$ .

STEP 13

Confirm that the fourth polynomial is not in Standard Form.
Since the terms are not in descending order by degree (the $x^{3}$ term should come before the $x^{2}$ and $x$ terms), the fourth polynomial is not in Standard Form.

STEP 14

List all polynomials that are in Standard Form.
The polynomials in Standard Form are: $5 x^{5}+4 x^{4}+12 x^{3}-23 x+15$ $2 x^{4}-12 x^{2}+5 x-9$

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Find the polynomials in standard form. Choose all correct options. 5x5+4x4+12x3−23x+155x^5 + 4x^4 + 12x^3 - 23x + 155x5+4x4+12x3−23x+15, 2x4−12x2+5x−92x^4 - 12x^2 + 5x - 92x4−12x2+5x−9, 5−3x5 - 3x5−3x, 4x2+12x−13x34x^2 + 12x - 13x^34x2+12x−13x3.

STEP 1

Assumptions1. We are looking for polynomials in Standard Form.2. Standard Form for a polynomial means the terms are written in descending order of their degree (the highest power of x comes first, followed by lower powers).3. The coefficients of the terms can be any real number.

STEP 2

Examine the first polynomial to determine if it is in Standard Form.5x5+4x4+12x3−23x+155 x^{5}+4 x^{4}+12 x^{3}-23 x+155x5+4x4+12x3−23x+15

STEP 3

Check if the terms of the first polynomial are in descending order by degree.The terms are in order: x5x^{5}x5, x4x^{4}x4, x3x^{3}x3, x1x^{1}x1, and the constant term.

STEP 4

Confirm that the first polynomial is in Standard Form.Since the terms are in descending order by degree, the first polynomial is in Standard Form.

STEP 5

Examine the second polynomial to determine if it is in Standard Form.2x4−12x2+5x−92 x^{4}-12 x^{2}+5 x-92x4−12x2+5x−9

STEP 6

Check if the terms of the second polynomial are in descending order by degree.The terms are in order: x4x^{4}x4, x2x^{2}x2, x1x^{1}x1, and the constant term.

STEP 7

Confirm that the second polynomial is in Standard Form.Since the terms are in descending order by degree, the second polynomial is in Standard Form.

STEP 8

Examine the third polynomial to determine if it is in Standard Form.5−3x5-3 x5−3x

STEP 9

Check if the terms of the third polynomial are in descending order by degree.The terms are in order: the constant term, and then x1x^{1}x1.

STEP 10

Confirm that the third polynomial is not in Standard Form.Since the terms are not in descending order by degree (the constant term should come after the xxx term), the third polynomial is not in Standard Form.

STEP 11

Examine the fourth polynomial to determine if it is in Standard Form.4x2+12x−13x34 x^{2}+12 x-13 x^{3}4x2+12x−13x3

STEP 12

Check if the terms of the fourth polynomial are in descending order by degree.The terms are in order: x2x^{2}x2, x1x^{1}x1, and then x3x^{3}x3.

STEP 13

Confirm that the fourth polynomial is not in Standard Form.Since the terms are not in descending order by degree (the x3x^{3}x3 term should come before the x2x^{2}x2 and xxx terms), the fourth polynomial is not in Standard Form.

STEP 14

List all polynomials that are in Standard Form.The polynomials in Standard Form are: 5x5+4x4+12x3−23x+155 x^{5}+4 x^{4}+12 x^{3}-23 x+155x5+4x4+12x3−23x+15 2x4−12x2+5x−92 x^{4}-12 x^{2}+5 x-92x4−12x2+5x−9

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Find the polynomials in standard form. Choose all correct options. $5x^5 + 4x^4 + 12x^3 - 23x + 15$ , $2x^4 - 12x^2 + 5x - 9$ , $5 - 3x$ , $4x^2 + 12x - 13x^3$ .

Assumptions
1. We are looking for polynomials in Standard Form.
2. Standard Form for a polynomial means the terms are written in descending order of their degree (the highest power of x comes first, followed by lower powers).
3. The coefficients of the terms can be any real number.

Examine the first polynomial to determine if it is in Standard Form.
$5 x^{5}+4 x^{4}+12 x^{3}-23 x+15$

Check if the terms of the first polynomial are in descending order by degree.
The terms are in order: $x^{5}$ , $x^{4}$ , $x^{3}$ , $x^{1}$ , and the constant term.

Confirm that the first polynomial is in Standard Form.
Since the terms are in descending order by degree, the first polynomial is in Standard Form.

Examine the second polynomial to determine if it is in Standard Form.
$2 x^{4}-12 x^{2}+5 x-9$

Check if the terms of the second polynomial are in descending order by degree.
The terms are in order: $x^{4}$ , $x^{2}$ , $x^{1}$ , and the constant term.

Confirm that the second polynomial is in Standard Form.
Since the terms are in descending order by degree, the second polynomial is in Standard Form.

Examine the third polynomial to determine if it is in Standard Form.
$5-3 x$

Check if the terms of the third polynomial are in descending order by degree.
The terms are in order: the constant term, and then $x^{1}$ .

Confirm that the third polynomial is not in Standard Form.
Since the terms are not in descending order by degree (the constant term should come after the $x$ term), the third polynomial is not in Standard Form.

Examine the fourth polynomial to determine if it is in Standard Form.
$4 x^{2}+12 x-13 x^{3}$

Check if the terms of the fourth polynomial are in descending order by degree.
The terms are in order: $x^{2}$ , $x^{1}$ , and then $x^{3}$ .

Confirm that the fourth polynomial is not in Standard Form.
Since the terms are not in descending order by degree (the $x^{3}$ term should come before the $x^{2}$ and $x$ terms), the fourth polynomial is not in Standard Form.

List all polynomials that are in Standard Form.
The polynomials in Standard Form are: $5 x^{5}+4 x^{4}+12 x^{3}-23 x+15$ $2 x^{4}-12 x^{2}+5 x-9$