Polynomials and Polynomial Functions

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, P(x) = 4x^3 - 3x^2 + 2x - 1 is a polynomial.

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial. For example, in P(x) = 3x^4 - 7x^2 - 2x^7 - x + 4, the degree is 7.

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How do you find the difference of two polynomials?

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To find the difference of two polynomials, subtract the corresponding coefficients of like terms. For example, (4x^3 - 3x^2 + 6x - 4) - (-2x^3 + x^2 - 20) results in 6x^3 - 4x^2 + 6x + 16.

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What is factoring in polynomials?

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Factoring a polynomial involves expressing it as a product of its factors. For example, x^3 + 1 can be factored as (x + 1)(x^2 - x + 1).

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What is the difference of cubes formula?

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The difference of cubes formula states that a^3 - b^3 = (a - b)(a^2 + ab + b^2). For example, 8x^3 - 1 can be factored using this formula.

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What is the sum of cubes formula?

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The sum of cubes formula states that a^3 + b^3 = (a + b)(a^2 - ab + b^2). This is useful for factoring expressions like x^3 + 1.

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How do you simplify polynomial expressions?

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To simplify polynomial expressions, combine like terms by adding or subtracting their coefficients. For example, (7x^2 - 16x - 13) - (-4 + 12x - 8x^2) simplifies to 15x^2 - 28x - 9.

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What are like terms in polynomials?

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Like terms are terms that have the same variable raised to the same power. For example, 3x^2 and -5x^2 are like terms.

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What is a monomial?

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A monomial is a polynomial with only one term. For example, 5x^3 is a monomial.

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What is a binomial?

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A binomial is a polynomial with two terms. For example, 3x^2 + 2x is a binomial.