Questions and Answers
What is a monomial?
How do you find the degree of the monomial -5a^3b^6?
Add the exponents: 3 + 6 = 9
What defines a polynomial?
What is the degree of a polynomial?
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Find the degree of the polynomial 5x^2 - 18x^5.
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What is standard form of a polynomial?
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What is the leading coefficient in the polynomial written in standard form?
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Write the polynomial 20x - 4x^3 + 2 - x^2 in standard form.
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Name the polynomial with the degree of 0.
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Name a polynomial with the degree of 1.
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Name a polynomial with the degree of 2.
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Name a polynomial with the degree of 3.
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Name a polynomial with the degree of 4.
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Name a polynomial with the degree of 5 or more.
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What do you call a term that has 1 part?
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What do you call a term that has 2 parts?
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What do you call a term that has 3 parts?
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What do you call a term that has 4 or more parts?
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Classify the polynomial y^2 + y + 4 according to its degree and number of terms.
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Determine which groups are alike: -3, -3x, 4x, 5x^2, 3x^3.
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Simplify the expression 4x + 2x.
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Simplify the expression 5n + 6n^2.
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Add the polynomials (2x^2 + x - 5) + (x^2 + x + 6).
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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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Test your understanding of monomials and polynomials with these flashcards. Learn the definitions, how to determine degrees, and more to solidify your knowledge in algebra. Perfect for students wanting to master algebraic concepts.
