Factoring Polynomials with Leading Coefficient of 1

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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 whole number powers and coefficients. For example, 2x^2 + 3x + 1 is a polynomial.

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What does it mean to factor a polynomial?

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Factoring a polynomial means expressing it as a product of its factors, which are polynomials of lower degrees. For example, factoring x^2 - 5x + 6 gives (x - 2)(x - 3).

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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 the polynomial 3x^2 + 2x + 1, the leading coefficient is 3.

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What is the standard form of a polynomial?

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The standard form of a polynomial is written in descending order of the degree of its terms. For example, 4x^3 + 2x^2 - x + 5 is in standard form.

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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), 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 (x - 2)(x + 3) = 0, then x - 2 = 0 or x + 3 = 0.

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How do you factor a quadratic polynomial of the form x^2 + bx + c?

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To factor a quadratic polynomial of the form x^2 + bx + c, find two numbers that multiply to c and add to b. For example, for x^2 + 5x + 6, the numbers are 2 and 3, so it factors to (x + 2)(x + 3).

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What is the significance of the constant term in a polynomial?

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The constant term in a polynomial is the term without a variable. It affects the y-intercept of the graph of the polynomial function.

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What is the relationship between the roots of a polynomial and its factors?

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The roots of a polynomial are the values of the variable that make the polynomial equal to zero. Each root corresponds to a factor of the polynomial.

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How can you check if your factorization is correct?

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You can check if your factorization is correct by multiplying the factors back together to see if you obtain the original polynomial.