Polynomial Division

Front

What is polynomial division?

Back

Polynomial division is the process of dividing a polynomial by another polynomial of the same or lower degree, similar to long division with numbers.

Front

What is the Remainder Theorem?

Back

The Remainder Theorem states that the remainder of the division of a polynomial p(x) by a linear divisor (x - c) is equal to p(c).

Front

How do you evaluate p(2) for p(x) = 9x^4 - 45x^3 + 37x^2 + x + 2?

Back

To evaluate p(2), substitute 2 into the polynomial: p(2) = 9(2)^4 - 45(2)^3 + 37(2)^2 + 2 + 2 = -64.

Front

What is synthetic division?

Back

Synthetic division is a simplified form of polynomial division that is used when dividing by a linear factor of the form (x - c).

Front

What is the result of dividing (5x^3 + 36x^2 + 45x + 58) by (x + 6)?

Back

The result is 5x^2 + 6x + 9 with a remainder of 4.

Front

What does it mean for (x - 4) to be a factor of a polynomial?

Back

If (x - 4) is a factor of a polynomial, then the polynomial evaluates to zero when x = 4.

Front

How do you determine if (x - 4) is a factor of (x^3 + x^2 - 16x - 16)?

Back

Use the Factor Theorem: evaluate the polynomial at x = 4. If p(4) = 0, then (x - 4) is a factor.

Front

What is the significance of the coefficients in polynomial division?

Back

The coefficients represent the values of the polynomial at different degrees, and their order affects the polynomial's behavior and roots.

Front

What is the quotient in polynomial division?

Back

The quotient is the result of the division process, representing how many times the divisor fits into the dividend.

Front

What is a binomial?

Back

A binomial is a polynomial with exactly two terms, such as (x + 2) or (3x - 5).