Flashcard 1-1: Polynomials, Factoring, and Rational Expressions

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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. It can be expressed in the form: $$a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $$a_n, a_{n-1}, ..., a_0$$ are constants.

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What does it mean to factor a polynomial?

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Factoring a polynomial involves expressing it as a product of its factors, which are polynomials of lower degrees. For example, the polynomial $$x^2 - 5x + 6$$ can be factored as $$(x - 2)(x - 3)$$.

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What is the difference of squares?

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The difference of squares is a special factoring pattern that states: $$a^2 - b^2 = (a - b)(a + b)$$. For example, $$x^2 - 9 = (x - 3)(x + 3)$$.

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What is the greatest common factor (GCF)?

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The greatest common factor (GCF) of two or more terms is the largest factor that divides all the terms. For example, the GCF of $$6x^2$$ and $$9x$$ is $$3x$$.

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How do you simplify rational expressions?

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To simplify a rational expression, factor the numerator and denominator and then cancel any common factors. For example, $$\frac{2x^2}{4x} = \frac{2x}{4} = \frac{x}{2}$$.

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What is a rational expression?

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A rational expression is a fraction where the numerator and the denominator are both polynomials. For example, $$\frac{x^2 + 2x + 1}{x - 1}$$ is a rational expression.

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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.

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What is the power of a product rule?

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The power of a product rule states that $$(ab)^n = a^n b^n$$. For example, $$(2x)^3 = 2^3 x^3 = 8x^3$$.

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What is the power of a quotient rule?

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The power of a quotient rule states that $$(\frac{a}{b})^n = \frac{a^n}{b^n}$$. For example, $$(\frac{2}{3})^2 = \frac{2^2}{3^2} = \frac{4}{9}$$.

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What is the zero product property?

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The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. For example, if $$x(x - 3) = 0$$, then $$x = 0$$ or $$x - 3 = 0$$.