Remainder Theorem

Front

What is the Remainder Theorem?

Back

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

Front

How do you find P(4) for the polynomial P(x) = x^3 + x^2 - x - 9?

Back

To find P(4), substitute 4 into the polynomial: P(4) = 4^3 + 4^2 - 4 - 9 = 67.

Front

What is the remainder when x^4 - 5 is divided by x + 3?

Back

Using the Remainder Theorem, substitute -3 into the polynomial: P(-3) = (-3)^4 - 5 = 76.

Front

How do you calculate the remainder of (4x^3 - 3x^2 - 4x + 2) ÷ (x - 3)?

Back

Substitute 3 into the polynomial: P(3) = 4(3)^3 - 3(3)^2 - 4(3) + 2 = 71.

Front

What is the remainder when n^3 - 7 is divided by n + 5?

Back

Substitute -5 into the polynomial: P(-5) = (-5)^3 - 7 = -132.

Front

What is the significance of the Remainder Theorem in polynomial division?

Back

It allows for quick calculation of the remainder without performing long division.

Front

If P(x) = x^2 + 2x + 1, what is P(-1)?

Back

P(-1) = (-1)^2 + 2(-1) + 1 = 0.

Front

What is the relationship between the Remainder Theorem and synthetic division?

Back

Synthetic division can be used to find the remainder of a polynomial when divided by a linear factor (x - c).

Front

How can you verify the Remainder Theorem?

Back

By performing polynomial long division and comparing the calculated remainder with P(c).

Front

What is the remainder when x^3 + 2x^2 - 5 is divided by x - 1?

Back

P(1) = 1^3 + 2(1)^2 - 5 = -2.