Understanding Polynomial Zeros and Multiplicity

The video tutorial introduces the concept of zeros in polynomials, starting with linear polynomials and progressing to higher order polynomials. It explains the importance of zeros, how to solve for them, and introduces the concept of multiplicity. The tutorial covers various techniques for solving polynomials, including factoring and substitution methods, and demonstrates how to determine zeros and their multiplicity.

What is the primary focus when discussing the zeros of a polynomial?

• Calculating the derivative of the polynomial

• Determining when the polynomial equals zero

• Identifying the degree of the polynomial

• Finding the maximum value of the polynomial

How do you find the zero of a linear polynomial like f(x) = x + 3?

• Set f(x) to 1

• Set x to 0

• Set f(x) to 0

• Set x to 1

What happens to the graph of a polynomial at a zero with odd multiplicity?

• The graph remains below the x-axis

• The graph remains above the x-axis

• The graph crosses the x-axis

• The graph touches the x-axis but does not cross it

Which of the following is true about even multiplicity?

• The graph is always increasing

• The graph has no x-intercepts

• The graph touches the x-axis but does not cross it

• The graph crosses the x-axis

What is the first step in solving a polynomial equation by factoring?

• Find the derivative of the polynomial

• Set the polynomial equal to one

• Set the polynomial equal to zero

• Multiply all terms by a constant

In the substitution method, what is the purpose of substituting x^4 with x^2?

• To simplify the polynomial for easier factoring

• To increase the degree of the polynomial

• To eliminate the constant term

• To find the derivative

Which two numbers multiply to -6 and add to 1 in the trinomial x^2 + x - 6?

• 1 and -6

• 6 and -1

• 3 and -2

• 2 and -3

What are the zeros of the polynomial x(x + 3)(x - 2)?

• x = 0, x = 3, x = -2

• x = 0, x = -3, x = 2

• x = 0, x = 2, x = -3

• x = 3, x = -3, x = 2

How can you determine the multiplicity of a zero in a factored polynomial?

• By the coefficient of the term

• By the degree of the polynomial

• By the constant term

• By the exponent of the factor

What is the significance of a zero with a multiplicity of one?

• The graph is always decreasing

• The graph has no x-intercepts

• The graph crosses the x-axis

• The graph touches the x-axis but does not cross it