Factor Theorem

The video tutorial covers the concept of divisibility and polynomial factorization, starting with a simple example of distributing pencils and candies. It introduces the remainder and factor theorems, providing proofs and applications. The tutorial explains how to determine if a polynomial is a factor of another without division, using the factor theorem. It also covers techniques for factorizing quadratic polynomials and provides step-by-step guidance on polynomial factorization. Advanced examples are included to illustrate the application of these concepts.

What is the main idea behind Sam's method of distributing candies and pencils?

• To distribute items based on their favorite colors

• To keep some items for himself

• To give each friend an equal share

• To ensure each friend gets a different number of items

According to the Remainder Theorem, what is the remainder when a polynomial P(x) is divided by a linear polynomial X - A?

• X

• P(A)

• 0

• A

What does the Factor Theorem state about a polynomial P(x) and a real number a?

• X - A is a factor if P(a) is not zero

• X - A is a factor if P(a) is zero

• X + A is a factor if P(a) is not zero

• X + A is a factor if P(a) is zero

What is the converse of the Factor Theorem?

• If X - A is a factor, then P(a) is zero

• If X + A is a factor, then P(a) is zero

• If X - A is not a factor, then P(a) is zero

• If X + A is not a factor, then P(a) is zero

In the method of splitting the middle term, what is the goal?

• To eliminate the middle term

• To divide the polynomial by the middle term

• To split the middle term into two terms whose sum is the middle term and product is the constant term

• To find the sum and product of the middle term

When factorizing a quadratic polynomial using the middle term method, what should the product of the two new terms equal?

• The leading coefficient

• The constant term

• The middle term

• The sum of all coefficients

What is the first step in factorizing a polynomial using the Factor Theorem?

• Find all possible factors of the leading coefficient

• Find all possible factors of the constant term

• Divide the polynomial by its degree

• Multiply the polynomial by its degree

How do you determine if a value is a zero of a polynomial?

• If the polynomial equals the constant term when the value is substituted

• If the polynomial equals the degree when the value is substituted

• If the polynomial equals one when the value is substituted

• If the polynomial equals zero when the value is substituted

What is the relationship between the degree of a polynomial and the number of its linear factors?

• The number of linear factors is always one more than the degree

• The number of linear factors is equal to the degree

• The number of linear factors is always one less than the degree

• The number of linear factors is half the degree

What is the simplified form of the polynomial X^3 + 2X^2 - 19X + 30 given that X + 6 is a factor?

• X + 6 and X^2 - 4X + 5

• X - 6 and X^2 + 4X - 5

• X - 6 and X^2 - 4X + 5

• X + 6 and X^2 + 4X - 5