Remainder Theorem & Factor 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

What is the Factor Theorem?

Back

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

Front

How do you determine if (x - c) is a factor of P(x)?

Back

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

Front

If P(3) = 0, what can be concluded about P(x)?

Back

If P(3) = 0, then (x - 3) is a factor of P(x).

Front

What is the value of P(2) for P(x) = 2x^4 + x^3 - 3x^2 - x - 15?

Back

P(2) = 11.

Front

Is (x - 3) a factor of P(x) = x^3 - 3x^2 + 2x + 2?

Back

No, (x - 3) is not a factor because P(3) ≠ 0.

Front

What does it mean if P(c) = 0?

Back

If P(c) = 0, then c is a root of the polynomial P(x).

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How can you find the roots of a polynomial?

Back

To find the roots of a polynomial, use the Factor Theorem and synthetic division or polynomial long division.

Front

What is synthetic division?

Back

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor.

Front

What is the significance of the leading coefficient in a polynomial?

Back

The leading coefficient determines the end behavior of the polynomial function.