Remainder Theorem & Factor Theorem

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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).

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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.

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How do you find the remainder of a polynomial when divided by a linear factor?

Back

To find the remainder of a polynomial f(x) when divided by (x - c), evaluate f(c).

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

Back

The Remainder Theorem allows us to quickly find the remainder without performing full polynomial long division.

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If P(x) = 2x^4 + x^3 - 3x^2 - x - 15, what is P(2)?

Back

P(2) = 11.

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Is (x - 3) a factor of the polynomial x^3 - 3x^2 + 2x + 2?

Back

No, (x - 3) is not a factor.

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What is the result of dividing 3x^3 - 5x^2 - 4x + 1 by (x + 3)?

Back

The remainder is -113.

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What does it mean if f(c) = 0 for a polynomial f(x)?

Back

It means that (x - c) is a factor of f(x).

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How can you determine if a polynomial has a specific factor?

Back

Evaluate the polynomial at the root of the factor; if the result is zero, the factor exists.

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What is synthetic division?

Back

Synthetic division is a simplified form of polynomial division that is used when dividing by linear factors.