Podcast
Questions and Answers
What is a monomial?
What is a monomial?
- A sum of numbers and variables
- A product of numbers and variables with positive whole number exponents (correct)
- A number only
- None of the above
How do you find the degree of the monomial -5a^3b^6?
How do you find the degree of the monomial -5a^3b^6?
Add the exponents: 3 + 6 = 9
What defines a polynomial?
What defines a polynomial?
- A single number
- A product of variables
- One monomial or a sum/difference of monomials (correct)
- Only a binomial
What is the degree of a polynomial?
What is the degree of a polynomial?
Find the degree of the polynomial 5x^2 - 18x^5.
Find the degree of the polynomial 5x^2 - 18x^5.
What is standard form of a polynomial?
What is standard form of a polynomial?
What is the leading coefficient in the polynomial written in standard form?
What is the leading coefficient in the polynomial written in standard form?
Write the polynomial 20x - 4x^3 + 2 - x^2 in standard form.
Write the polynomial 20x - 4x^3 + 2 - x^2 in standard form.
Name the polynomial with the degree of 0.
Name the polynomial with the degree of 0.
Name a polynomial with the degree of 1.
Name a polynomial with the degree of 1.
Name a polynomial with the degree of 2.
Name a polynomial with the degree of 2.
Name a polynomial with the degree of 3.
Name a polynomial with the degree of 3.
Name a polynomial with the degree of 4.
Name a polynomial with the degree of 4.
Name a polynomial with the degree of 5 or more.
Name a polynomial with the degree of 5 or more.
What do you call a term that has 1 part?
What do you call a term that has 1 part?
What do you call a term that has 2 parts?
What do you call a term that has 2 parts?
What do you call a term that has 3 parts?
What do you call a term that has 3 parts?
What do you call a term that has 4 or more parts?
What do you call a term that has 4 or more parts?
Classify the polynomial y^2 + y + 4 according to its degree and number of terms.
Classify the polynomial y^2 + y + 4 according to its degree and number of terms.
Determine which groups are alike: -3, -3x, 4x, 5x^2, 3x^3.
Determine which groups are alike: -3, -3x, 4x, 5x^2, 3x^3.
Simplify the expression 4x + 2x.
Simplify the expression 4x + 2x.
Simplify the expression 5n + 6n^2.
Simplify the expression 5n + 6n^2.
Add the polynomials (2x^2 + x - 5) + (x^2 + x + 6).
Add the polynomials (2x^2 + x - 5) + (x^2 + x + 6).
Subtract the polynomials (2x^2 - x - 4) - (3x^2 - 5x + 3).
Subtract the polynomials (2x^2 - x - 4) - (3x^2 - 5x + 3).
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Study Notes
Monomials
- A monomial consists of a number, variable, or a product of numbers and variables with positive integer exponents.
- The degree of a monomial is determined by the sum of its exponents. For example, in -5a^3b^6, the degree is 3 + 6 = 9.
Polynomials
- A polynomial can be a single monomial or a sum/difference of monomials.
- The degree of a polynomial is defined by the term with the highest degree.
Degree of Polynomials
- To find the degree of a polynomial like 5x^2 - 18x^5, identify the term with the greatest degree, which is 5 in this case.
- Polynomials are written in standard form when the terms are in descending order based on their degree.
Leading Coefficient
- In standard form, the leading coefficient is the coefficient of the term with the highest degree.
Writing in Standard Form
- Example: The polynomial 20x - 4x^3 + 2 - x^2 is rewritten as -4x^3 - x^2 + 20x + 2, with a leading coefficient of -4.
Classification of Polynomials
- Constant (degree 0): e.g., y = 2.
- Linear (degree 1): e.g., y = x + 2.
- Quadratic (degree 2): e.g., y = x^2 + 2x + 3.
- Cubic (degree 3): e.g., y = x^3 + 7.
- Quartic (degree 4): e.g., y = x^4 + 2x^2 + 1.
- Degree of 5 or more: denoted as 5th, 6th, etc.
Types of Terms
- Monomial: a term with one part.
- Binomial: a term with two parts.
- Trinomial: a term with three parts.
- Polynomial: a term with four or more parts.
Classifying Polynomials
- For the polynomial y^2 + y + 4:
- Degree: 2 (quadratic).
- Number of terms: 3 (trinomial).
Identifying Like Terms
- From the group -3, -3x, 4x, 5x^2, 3x^3, the like terms are -3x and 4x because they both contain x to the first power.
Simplifying Expressions
- For expressions like 4x + 2x, simplify to 6x.
- In 5n + 6n^2, the expression remains as is since there are no like terms.
Polynomial Operations
- Adding polynomials: (2x^2 + x - 5) + (x^2 + x + 6) results in 3x^2 + 2x + 1.
- Subtracting polynomials: (2x^2 - x - 4) - (3x^2 - 5x + 3) results in -1x^2 + 4x - 7.
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