The 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 - k), the remainder of this division is f(k).

Front

What is the Factor Theorem?

Back

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

Front

How do you determine if k is a zero of a polynomial function?

Back

To determine if k is a zero of a polynomial function f(x), evaluate f(k). If f(k) = 0, then k is a zero.

Front

What is the significance of a zero in a polynomial function?

Back

A zero of a polynomial function is a value of x that makes the function equal to zero, indicating that the graph of the function intersects the x-axis.

Front

If f(x) = 3x^2 - 9x + 2, what is f(-3)?

Back

f(-3) = 3(-3)^2 - 9(-3) + 2 = 27 + 27 + 2 = 56.

Front

What is the result of dividing 2x^3 + 5x^2 + 9 by x + 3?

Back

The result is 2x^2 - x + 3.

Front

What does it mean if the remainder is 0 when dividing a polynomial by (x - k)?

Back

It means that k is a root of the polynomial, and (x - k) is a factor of the polynomial.

Front

How can you find the remainder of a polynomial division without performing long division?

Back

You can find the remainder by evaluating the polynomial at the value of k using the Remainder Theorem.

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 - k).

Front

If f(x) = 4x^2 - 33x - 27, what is f(9)?

Back

f(9) = 4(9)^2 - 33(9) - 27 = 324 - 297 - 27 = 0.