Polynomials End Behavior & Division

Front

What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, 2x^3 - 5x + 4 is a polynomial.

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What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial 3x^4 + 2x^2 - 5, the degree is 4.

Front

What is the leading coefficient of a polynomial?

Back

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 does the end behavior of a polynomial describe?

Back

The end behavior of a polynomial describes how the values of the polynomial behave as the input (x) approaches positive or negative infinity.

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How do you determine the end behavior of a polynomial?

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The end behavior can be determined by the degree and the leading coefficient of the polynomial. If the degree is even and the leading coefficient is positive, both ends go up; if negative, both ends go down.

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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 divisor of the form (x - c). It is faster than long division and uses coefficients.

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What is the purpose of a placeholder in polynomial division?

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A placeholder is used to ensure that all degrees of the polynomial are represented, which helps in maintaining the correct structure during division.

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What is the significance of the quadratic term in polynomial division?

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The quadratic term is significant because it helps in determining the necessary placeholders and the structure of the polynomial during division.

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How do you identify the leading coefficient and degree from a graph?

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From a graph, the leading coefficient can be inferred from the direction of the ends of the graph, and the degree can be determined by the number of times the graph crosses the x-axis.

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What does it mean if a polynomial has a negative leading coefficient?

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If a polynomial has a negative leading coefficient, the end behavior will be such that the left end of the graph goes up and the right end goes down if the degree is odd.