PCX Chapter 2 FE Review

Front

What are the possible rational zeros for a polynomial function?

Back

The possible rational zeros can be found using 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.

Front

Define 'multiplicity' in the context of polynomial roots.

Back

Multiplicity refers to the number of times a particular root appears in a polynomial. For example, if (x - r) is a factor of a polynomial with an exponent of n, then r is a root with multiplicity n.

Front

How do you determine the multiplicity of a root from a polynomial's graph?

Back

The multiplicity of a root can be determined by observing the behavior of the graph at the root. If the graph touches the x-axis and turns around, the root has an even multiplicity. If it crosses the x-axis, the root has an odd multiplicity.

Front

What is synthetic division and when is it used?

Back

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor of the form (x - c). It is faster and requires less writing than traditional long division.

Front

What is the significance of the remainder in synthetic division?

Back

The remainder in synthetic division indicates whether the divisor (x - c) is a factor of the polynomial. If the remainder is zero, then (x - c) is a factor.

Front

How do you find the roots of a polynomial function?

Back

To find the roots of a polynomial function, set the polynomial equal to zero and solve for x. This can involve factoring, using the quadratic formula, or applying synthetic division.

Front

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

Back

A root of multiplicity 2 means that the root is repeated twice. This affects the graph, causing it to touch the x-axis at that root but not cross it.

Front

Explain the Rational Root Theorem.

Back

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

Front

What is the end behavior of a polynomial function?

Back

The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

Front

How do you identify the leading coefficient of a polynomial?

Back

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. For example, in the polynomial 3x^4 + 2x^3 - x + 5, the leading coefficient is 3.