A2 - Factor and Remainder Theorem

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What is the Factor Theorem?

Back

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0.

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What is the Remainder Theorem?

Back

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

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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 method of dividing a polynomial by a linear factor of the form (x - c).

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How do you use synthetic division to evaluate f(c)?

Back

To evaluate f(c) using synthetic division, set up the synthetic division with c and the coefficients of f(x). The remainder will be f(c).

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What is a zero of a polynomial function?

Back

A zero of a polynomial function f(x) is a value of x for which f(x) = 0.

Front

How can you find the real zeros of a polynomial?

Back

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

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What does it mean for a polynomial to be factored completely?

Back

A polynomial is factored completely when it is expressed as a product of irreducible polynomials over the real numbers.

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What is the significance of the degree of a polynomial?

Back

The degree of a polynomial indicates the highest power of the variable and determines the polynomial's end behavior and the maximum number of real zeros.

Front

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

Back

The factors of a polynomial correspond to its zeros; if (x - c) is a factor, then c is a zero of the polynomial.