Jan. 2 Polynomial Functions & sum/diff. cubes

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What is a polynomial function?

Back

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer.

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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 the polynomial $$2x^3 + 3x^2 + 4$$, the degree is 3.

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How do you factor a sum of cubes?

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The sum of cubes can be factored using the formula: $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$. For example, $$x^3 + 8 = (x + 2)(x^2 - 2x + 4)$$.

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What is a cubic polynomial?

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A cubic polynomial is a polynomial of degree 3, which can be expressed in the form: $$f(x) = ax^3 + bx^2 + cx + d$$ where \(a \neq 0\).

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What is the difference between a monomial, binomial, and trinomial?

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A monomial has one term (e.g., $$3x^2$$), a binomial has two terms (e.g., $$x^2 + 4$$), and a trinomial has three terms (e.g., $$x^2 + 4x + 4$$).

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What is the formula for factoring a difference of cubes?

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The difference of cubes can be factored using the formula: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$. For example, $$27x^3 - 8 = (3x - 2)(9x^2 + 6x + 4)$$.

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How do you classify polynomials by degree?

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Polynomials can be classified as follows: 0 degree (constant), 1 degree (linear), 2 degree (quadratic), 3 degree (cubic), and so on.

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What is a quadratic polynomial?

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A quadratic polynomial is a polynomial of degree 2, expressed in the form: $$f(x) = ax^2 + bx + c$$ where \(a \neq 0\).

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What is the leading coefficient of a polynomial?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. For example, in $$4x^3 + 2x^2 + 3$$, the leading coefficient is 4.

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What is the standard form of a polynomial?

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The standard form of a polynomial is when the terms are arranged in descending order of their degrees. For example, $$3x^4 + 2x^3 - x + 5$$ is in standard form.