Factoring Polynomials, Division, Rational Zeros Thm.

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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, f(x) = 2x^3 - 4x^2 + 5x - 12 is a polynomial.

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What does it mean for a root to have a multiplicity of 2?

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A root has a multiplicity of 2 if it is a repeated root, meaning it is a solution to the polynomial equation that occurs twice. This affects the graph of the polynomial, causing it to touch the x-axis at that root but not cross it.

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How do you find the possible rational roots of a polynomial?

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Use the Rational Root Theorem, which states that any rational solution, expressed as a fraction p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.

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What is the Remainder Theorem?

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The Remainder Theorem states that if a polynomial f(x) is divided by (x - c), the remainder of that division is f(c). This helps in determining if (x - c) is a factor of f(x).

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What is the Factor Theorem?

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The Factor Theorem states that (x - c) is a factor of the polynomial f(x) if and only if f(c) = 0. This means c is a root of the polynomial.

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How do you determine the end behavior of a polynomial function?

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The end behavior of a polynomial function is determined by the leading term. If the leading coefficient is positive and the degree is even, the ends rise; if negative, the ends fall. If the degree is odd, the ends go in opposite directions.

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What is the significance of the leading coefficient in a polynomial?

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The leading coefficient affects the direction of the graph's end behavior and the overall shape of the polynomial function. It determines whether the graph rises or falls as x approaches positive or negative infinity.

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What is a rational zero?

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A rational zero is a solution to a polynomial equation that can be expressed as a fraction of two integers. The Rational Root Theorem helps identify potential rational zeros.

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How do you factor a polynomial?

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To factor a polynomial, look for common factors, use grouping, or apply special factoring techniques such as difference of squares, perfect square trinomials, or the sum/difference of cubes.

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form (x - c). It is faster than long division and is used to find the quotient and remainder.