2. Finding Zeros by Factoring

Front

What does it mean to find the zeros of a function?

Back

Finding the zeros of a function means determining the values of x for which the function f(x) equals zero. These values are also known as the roots of the function.

Front

What is the general form of a cubic function?

Back

A cubic function can be expressed in the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a ≠ 0.

Front

How can you factor a polynomial to find its zeros?

Back

To factor a polynomial, you can look for common factors, use grouping, or apply special factoring techniques such as the difference of squares or the sum/difference of cubes. Once factored, set each factor equal to zero to find the zeros.

Front

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

Back

The leading coefficient determines the end behavior of the polynomial function. If it is positive, the function will rise to the right; if negative, it will fall to the right.

Front

What is the relationship between the degree of a polynomial and the number of zeros it can have?

Back

The degree of a polynomial indicates the maximum number of zeros it can have. For example, a cubic polynomial (degree 3) can have up to 3 real zeros.

Front

What is synthetic division, and how is it used in finding zeros?

Back

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor. It can help find zeros by simplifying the polynomial and identifying potential roots.

Front

What is the Rational Root Theorem?

Back

The Rational Root Theorem states that any rational solution (or root) of a polynomial equation, in the form of p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient.

Front

How do you determine the multiplicity of a zero?

Back

The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. It can be determined by the exponent of the factor associated with that zero when the polynomial is factored.

Front

What is the difference between real and complex zeros?

Back

Real zeros are the x-values where the function crosses the x-axis, while complex zeros are solutions that involve imaginary numbers and do not correspond to x-intercepts on the graph.

Front

How can you verify if a value is a zero of a polynomial function?

Back

To verify if a value is a zero of a polynomial function, substitute the value into the function. If the result is zero, then the value is a zero of the function.