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 that 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 f(x)?

Back

To determine if (x - c) is a factor of 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 that is used specifically for dividing by linear factors of the form (x - c).

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

Back

f(2) = 2(2)^3 + 3(2)^2 - 5(2) + 6 = 16 + 12 - 10 + 6 = 24.

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

Back

The zeros of a polynomial are the values of x for which the polynomial equals zero, and each zero corresponds to a factor of the form (x - c).

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

Back

If f(c) = 0, it means that (x - c) is a factor of f(x) and c is a zero of the polynomial.

Front

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

Back

The leading coefficient determines the end behavior of the polynomial function as x approaches positive or negative infinity.

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What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression.