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 determine if (x - c) is a factor of a polynomial?

Back

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

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What is synthetic division?

Back

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

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If f(x) = x^3 + 2x^2 - 5x + 6, what is f(2)?

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f(2) = 2^3 + 2(2^2) - 5(2) + 6 = 8 + 8 - 10 + 6 = 12.

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

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The result is 2x^2 - x + 5 with a remainder of 9.

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What does it mean if a polynomial has a degree of 3?

Back

A polynomial of degree 3 is a cubic polynomial, which means its highest exponent is 3.

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

Back

The leading coefficient is the coefficient of the term with the highest degree and determines the end behavior of the polynomial graph.

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How can you find the zeros of a polynomial?

Back

To find the zeros of a polynomial, set the polynomial equal to zero and solve for x.

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What is the relationship between factors and zeros of a polynomial?

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The zeros of a polynomial correspond to the values of x that make the polynomial equal to zero, which are also the roots or factors of the polynomial.