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Answer :
Let's go through each polynomial and rewrite them in standard form. We'll identify the leading coefficient, degree, the number of terms, and name the polynomial.
### Problem 23: [tex]\(3x^3 + 2x^4 - 7x + x^2\)[/tex]
1. Standard Form: Arrange the terms in descending order of their exponents:
[tex]\(2x^4 + 3x^3 + x^2 - 7x\)[/tex]
2. Leading Coefficient: The coefficient of the term with the highest degree, which is [tex]\(2\)[/tex].
3. Degree: The highest exponent in the polynomial is [tex]\(4\)[/tex].
4. Number of Terms: There are 4 terms in the polynomial.
5. Name: This polynomial is called a "Polynomial".
### Problem 24: [tex]\(6x - 4x^4 + 5^7\)[/tex]
1. Standard Form: Simplify and arrange the terms:
[tex]\(-4x^4 + 6x + 78125\)[/tex] (where [tex]\(5^7 = 78125\)[/tex])
2. Leading Coefficient: The coefficient of the term with the highest degree, which is [tex]\(-4\)[/tex].
3. Degree: The highest exponent in the polynomial is [tex]\(4\)[/tex].
4. Number of Terms: There are 3 terms.
5. Name: This polynomial is called a "Trinomial".
### Problem 25: [tex]\(2x^3 + 10x - 9\)[/tex]
1. Standard Form: The terms are already arranged, so it's [tex]\(2x^3 + 10x - 9\)[/tex].
2. Leading Coefficient: The coefficient of the highest degree term, which is [tex]\(2\)[/tex].
3. Degree: The highest exponent in the polynomial is [tex]\(3\)[/tex].
4. Number of Terms: There are 3 terms.
5. Name: This polynomial is a "Trinomial".
### Problem 26: [tex]\(3x^2 + 2x^6 - 4x^4 - 1\)[/tex]
1. Standard Form: Arrange the terms in descending order of exponents:
[tex]\(2x^6 - 4x^4 + 3x^2 - 1\)[/tex]
2. Leading Coefficient: The coefficient of the highest degree term, which is [tex]\(2\)[/tex].
3. Degree: The highest exponent in the polynomial is [tex]\(6\)[/tex].
4. Number of Terms: There are 4 terms.
5. Name: This polynomial is a "Polynomial".
These steps help standardize and describe each polynomial in terms of mathematical properties like leading coefficient and degree, and it categorizes them based on the number of terms.
### Problem 23: [tex]\(3x^3 + 2x^4 - 7x + x^2\)[/tex]
1. Standard Form: Arrange the terms in descending order of their exponents:
[tex]\(2x^4 + 3x^3 + x^2 - 7x\)[/tex]
2. Leading Coefficient: The coefficient of the term with the highest degree, which is [tex]\(2\)[/tex].
3. Degree: The highest exponent in the polynomial is [tex]\(4\)[/tex].
4. Number of Terms: There are 4 terms in the polynomial.
5. Name: This polynomial is called a "Polynomial".
### Problem 24: [tex]\(6x - 4x^4 + 5^7\)[/tex]
1. Standard Form: Simplify and arrange the terms:
[tex]\(-4x^4 + 6x + 78125\)[/tex] (where [tex]\(5^7 = 78125\)[/tex])
2. Leading Coefficient: The coefficient of the term with the highest degree, which is [tex]\(-4\)[/tex].
3. Degree: The highest exponent in the polynomial is [tex]\(4\)[/tex].
4. Number of Terms: There are 3 terms.
5. Name: This polynomial is called a "Trinomial".
### Problem 25: [tex]\(2x^3 + 10x - 9\)[/tex]
1. Standard Form: The terms are already arranged, so it's [tex]\(2x^3 + 10x - 9\)[/tex].
2. Leading Coefficient: The coefficient of the highest degree term, which is [tex]\(2\)[/tex].
3. Degree: The highest exponent in the polynomial is [tex]\(3\)[/tex].
4. Number of Terms: There are 3 terms.
5. Name: This polynomial is a "Trinomial".
### Problem 26: [tex]\(3x^2 + 2x^6 - 4x^4 - 1\)[/tex]
1. Standard Form: Arrange the terms in descending order of exponents:
[tex]\(2x^6 - 4x^4 + 3x^2 - 1\)[/tex]
2. Leading Coefficient: The coefficient of the highest degree term, which is [tex]\(2\)[/tex].
3. Degree: The highest exponent in the polynomial is [tex]\(6\)[/tex].
4. Number of Terms: There are 4 terms.
5. Name: This polynomial is a "Polynomial".
These steps help standardize and describe each polynomial in terms of mathematical properties like leading coefficient and degree, and it categorizes them based on the number of terms.
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