Practice 4.7

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What is the Remainder Theorem?

Back

The Remainder Theorem states that the remainder of the division of a polynomial p(x) by (x - c) is equal to p(c).

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How do 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. This can involve factoring, using the quadratic formula, or synthetic division.

Front

What is synthetic division?

Back

Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form (x - c). It is faster than long division and is used primarily for polynomials.

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What are rational zeros?

Back

Rational zeros are potential solutions to a polynomial equation that can be expressed as a fraction p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

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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.

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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, evaluate the polynomial at c. If the result is zero, then (x - c) is a factor.

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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 in a polynomial. It affects the end behavior of the polynomial's graph.

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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. It determines the polynomial's behavior and the number of possible zeros.

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What is the relationship between the degree of a polynomial and the number of zeros?

Back

A polynomial of degree n can have at most n real zeros, counting multiplicities.

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What is the process of factoring a polynomial?

Back

Factoring a polynomial involves rewriting it as a product of its factors, which can include linear factors and irreducible polynomials.