Remainder and Factor Theorems

Front

What is the Remainder Theorem?

Back

The Remainder Theorem states that if a polynomial f(x) is divided by (x - c), the remainder of this division is f(c).

Front

What is the Factor Theorem?

Back

The Factor Theorem states that (x - c) is a factor of the polynomial f(x) if and only if f(c) = 0.

Front

How do you determine if a binomial is a factor of a polynomial?

Back

To determine if (x - c) is a factor of f(x), evaluate f(c). If f(c) = 0, then (x - c) is a factor.

Front

What is synthetic division?

Back

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear binomial of the form (x - c).

Front

What is the first step in synthetic division?

Back

The first step is to write down the coefficients of the polynomial in descending order of degree.

Front

What is the significance of the remainder in polynomial division?

Back

The remainder gives information about the polynomial's value at specific points and helps determine factors.

Front

If f(x) = 2x^3 - 3x + 5, what is f(1)?

Back

f(1) = 2(1)^3 - 3(1) + 5 = 4.

Front

What does it mean if the remainder is zero when dividing a polynomial?

Back

If the remainder is zero, it means that the divisor is a factor of the polynomial.

Front

How can you use the Remainder Theorem to evaluate polynomials quickly?

Back

You can substitute the value of c directly into the polynomial instead of performing long division.

Front

What is the result of dividing (x^2 - 4) by (x - 2)?

Back

The result is x + 2 with a remainder of 0, indicating (x - 2) is a factor.