Polynomial Factorization and Zeros

The video tutorial covers the process of finding rational roots using the p/q method, checking possible roots through division and the remainder theorem, and factoring cubic polynomials. It also discusses the use of the quadratic formula for non-factorable polynomials and introduces complex solutions. The tutorial concludes with an explanation of Descartes' rule of signs to determine the number of positive and negative real zeros.

What is the initial step in determining the potential rational zeros of a polynomial?

• List all factors of the middle term.

• List all factors of both the constant term and the leading coefficient.

• List all factors of the leading coefficient.

• List all factors of the constant term.

What is the purpose of listing all factors of the constant term and leading coefficient?

• To determine potential rational zeros.

• To graph the polynomial.

• To find the degree of the polynomial.

• To simplify the polynomial.

How can you test potential factors if you don't have a calculator?

• Use only positive factors.

• Use a graphing method to visualize the zeros.

• Start with positive one and then test negative one, then two, and so on.

• Start with the largest factor and work downwards.

What are two methods mentioned for checking if a number is a zero of a polynomial?

• Graphing and using a calculator.

• Synthetic division and the remainder theorem.

• Using the quadratic formula and graphing.

• Factoring and completing the square.

What does the remainder theorem state?

• The remainder is irrelevant in factorization.

• If the remainder is non-zero, the divisor is a factor.

• If the remainder is zero, the divisor is a factor.

• The remainder is always zero for any divisor.

What does a zero remainder in synthetic division indicate?

• The polynomial is quadratic.

• The polynomial is prime.

• The divisor is a factor.

• The divisor is not a factor.

What does the factor theorem state about a polynomial?

• The polynomial has no real roots.

• The polynomial is always factorable.

• If a polynomial is divisible by a binomial, the remainder is non-zero.

• If a polynomial is divisible by a binomial, the remainder is zero.

What is the purpose of using synthetic division in polynomial factorization?

• To find the degree of the polynomial.

• To graph the polynomial.

• To test potential zeros.

• To simplify the polynomial.

What does the factor theorem state about a polynomial?

• If a polynomial is divisible by a binomial, the remainder is zero.

• If a polynomial is divisible by a binomial, the remainder is non-zero.

• The polynomial is always factorable.

• The polynomial has no real roots.

What is the challenge with factoring a cubic polynomial?

• It may not be factorable using simple methods.

• It cannot be factored if it has no real roots.

• It always has complex roots.

• It always requires the quadratic formula.