Polynomial Equations and Factoring

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What is a polynomial?

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A polynomial is a mathematical expression that consists 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 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 $$3x^4 + 2x^3 - x + 5$$, the degree is 4.

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What does it mean to simplify a polynomial expression?

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Simplifying a polynomial expression involves combining like terms and reducing the expression to its simplest form. This may include adding, subtracting, or factoring terms.

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What are like terms in a polynomial?

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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, while $$3x^2$$ and $$4x^3$$ are not.

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What is the process of factoring a polynomial?

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Factoring a polynomial involves expressing it as a product of its factors. This can be done using methods such as grouping, the ac method, or using special products.

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What is the ac method for factoring?

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The ac method involves multiplying the leading coefficient (a) and the constant term (c) of a quadratic polynomial $$ax^2 + bx + c$$, then finding two numbers that multiply to ac and add to b.

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

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

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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 $$ab = 0$$, then either $$a = 0$$ or $$b = 0$$.

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How do you combine like terms in a polynomial?

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To combine like terms, add or subtract the coefficients of the terms that have the same variable and exponent. For example, $$2x + 3x = 5x$$.

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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, $$4x^3 + 2x^2 - x + 7$$ is in standard form.