Problem 13 Use the Rational Zero Theorem to... [FREE SOLUTION]

Chapter 2: Problem 13

Use the Rational Zero Theorem to list possible rational zeros for each polynomial function. $$P(x)=4 x^{4}-12 x^{3}-3 x^{2}+12 x-7$$

Short Answer

Expert verified

This polynomial's potential rational zeros are 卤1, 卤7, 卤1/2, 卤7/2, 卤1/4, 卤7/4.

Step by step solution

Understand the Rational Zero Theorem

The Rational Zero Theorem states that if the polynomial $ ax^n + bx^{n-1} + ... + Ax^1 + B $ has integer coefficients and $ p/q $ is a rational zero, then p is a factor of B and q is a factor of a. Here, a is the coefficient of the term with the highest power and B is the constant term. So, for our polynomial $ P(x) = 4x^{4}-12x^{3}-3x^{2}+12x-7 $, a = 4 and B = -7.

Identify Possible Values of p and q

Here, we will find all the factors of B (the constant term, -7) and a (coefficient of the highest power, 4). The factors of -7 are 卤1, 卤7 and factors of 4 are 卤1, 卤2, 卤4. These are our possible values of p and q.

Apply the Rational Zero Theorem

The Rational Zero Theorem states that the rational zeros of P(x) are all fractions of the form p/q, where p is a factor of B and q is a factor of a. So, we generate the possible rational zeros by taking every combination of p divided by q: 卤1/1, 卤7/1, 卤1/2, 卤7/2, 卤1/4, 卤7/4. Hence, our potential rational zeros are 卤1, 卤7, 卤1/2, 卤7/2, 卤1/4, 卤7/4.

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91影视

Use the Rational Zero Theorem to list possible rational zeros for each polynomial function. $$P(x)=4 x^{4}-12 x^{3}-3 x^{2}+12 x-7$$

Short Answer

Step by step solution

Understand the Rational Zero Theorem

Identify Possible Values of p and q

Apply the Rational Zero Theorem

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