Understanding the Factor Theorem

The video tutorial explains the factor theorem, which states that if a polynomial f(x) has a factor x - Z, then f(Z) = 0. This is closely related to the remainder theorem, which indicates that if plugging a number into a polynomial results in zero, that number is a root. The tutorial emphasizes the relationship between zeros and factors, demonstrating how to convert zeros into factors and solve polynomial equations. It concludes with a practical example and mentions alternative methods like long division.

What does the factor theorem state about a polynomial f(x) and a number z?

• If f(z) = 1, then x + z is a factor of f(x).

• If f(z) = 1, then x - z is a factor of f(x).

• If f(z) = 0, then x - z is a factor of f(x).

• If f(z) = 0, then x + z is a factor of f(x).

What does it mean if f(z) = 0 for a polynomial f(x)?

• z is a factor of f(x).

• x - z is a factor of f(x).

• x + z is a factor of f(x).

• z is not related to f(x).

How is the factor theorem similar to the remainder theorem?

• Both involve dividing polynomials by linear factors.

• Both state that if a number is a zero, it is also a factor.

• Both state that if a number is a zero, the remainder is zero.

• Both are used to find the degree of a polynomial.

How can you determine if a number is a zero of a polynomial?

• By checking if the polynomial equals zero when the number is substituted.

• By checking if the polynomial equals one when the number is substituted.

• By checking if the polynomial equals zero when the number is added.

• By checking if the polynomial equals one when the number is added.

What is the relationship between zeros and factors in a polynomial?

• Zeros are the same as factors.

• Zeros are the roots, and factors are the expressions that produce those roots.

• Factors are the roots, and zeros are the expressions that produce those roots.

• Zeros and factors are unrelated.

What is the zero product property used for in solving polynomials?

• To find the factors of the polynomial.

• To find the zeros of the polynomial.

• To find the coefficients of the polynomial.

• To find the degree of the polynomial.

What is the first step in solving a polynomial equation using the factor theorem?

• Find the zeros of the polynomial.

• Identify the degree of the polynomial.

• Write the polynomial in standard form.

• Factor the polynomial completely.

What is the relationship between the remainder theorem and the factor theorem?

• They both involve finding zeros and factors.

• They both involve finding the coefficients of a polynomial.

• They both involve finding the degree of a polynomial.

• They are unrelated.

In the example problem, what is the result of plugging in 1 into the function?

• The result is zero, indicating x - 1 is a factor.

• The result is not zero, indicating x + 1 is not a factor.

• The result is not zero, indicating x - 1 is not a factor.

• The result is zero, indicating x + 1 is a factor.

What is the significance of getting a zero remainder in polynomial division?

• It indicates the divisor is a factor.

• It indicates the polynomial is linear.

• It indicates the divisor is not a factor.

• It indicates the polynomial is prime.