An expression that is the sum of terms of the form \$a_nx^n\$, where \$a_n\$ is a real number and \$n\$ is a non-negative integer.
NoteThe degreeof a polynomial is the highest powerof the variable in the expression.
Properties of Polynomials
- Closure: The set of polynomials is closed under addition, subtraction, and multiplication. This means that the sum, difference, or product of two polynomials is also a polynomial.
- Commutative Property: The order of addition or multiplication does not affect the result.
- Associative Property: The grouping of terms does not affect the result.
- Distributive Property: Multiplication distributes over addition.
1. Verify the closure property by adding, subtracting, and multiplying two polynomials of your choice. 2. Demonstrate the commutative property with different polynomials. 3. Show the associative property by grouping terms in different ways. 4. Apply the distributive property to expand a product of polynomials.
TipWhen working with polynomials, always arrange terms in descending order of degree. This makes it easier to identify the degree and perform operations.
AnalogyThink of polynomials as a team of players (terms) working together. The degree is the captain, leading the team with the highest power. The properties ensure the team plays by the rules, maintaining order and consistency.
NoteTo add or subtract polynomials, align terms with the same degree and combine their coefficients. For multiplication, use the distributive property to expand the product.
NoteA polynomialis an expressionconsisting of variablesand coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
Self review1. Identify whether the following expressions are polynomials: 1. \$2x^3 - 4x + 7\$ 2. \$3x^{-1} + 5\$ 3. \$x^2 + 2\sqrt{x}\$ 2. Determine the degree of the polynomial \$4x^5 - 3x^3 + 2x^2 - x + 6\$.
Theory of KnowledgeHow do the properties of polynomials reflect the underlying structure of mathematics? Are these properties discovered or invented?
NoteStudents often confuse the degree of a polynomial with the number of terms. Remember, the degree is the highest power of the variable, not the count of terms.
TipWhen working with polynomials, always arrange terms in descending order of degree. This makes it easier to identify the degree and perform operations.
AnalogyThink of polynomials as a team of players (terms) working together. The degree is the captain, leading the team with the highest power. The properties ensure the team plays by the rules, maintaining order and consistency.
NoteTo add or subtract polynomials, align terms with the same degree and combine their coefficients. For multiplication, use the distributive property to expand the product.
Solution
