Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 74

Chapter 4, Problem 74

Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.

Verified step by step guidance

Step 1: Since 4x+3 is a factor of the polynomial, we can use the Factor Theorem which states that if a polynomial f(x) has a factor of the form x - a, then f(a) = 0. In this case, our factor is 4x + 3, so we can rewrite it in the form x - a by letting x = -3/4.

Step 2: Substitute x = -3/4 into the polynomial 20x^3+23x^2-10x+k. This will give us the equation 20(-3/4)^3 + 23(-3/4)^2 - 10(-3/4) + k = 0.

Step 3: Simplify the equation to find the value of k. Remember, the equation should equal zero because of the Factor Theorem.

Step 4: After simplifying, you will get an equation in the form k = some number.

Step 5: The number you get is the value of k that makes 4x+3 a factor of the polynomial 20x^3+23x^2-10x+k.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. In this context, determining if a polynomial, such as 20x^3 + 23x^2 - 10x + k, can be divided by another polynomial, like 4x + 3, without a remainder is essential. If 4x + 3 is a factor, then the polynomial can be rewritten in a simpler form, which is crucial for solving the problem.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is f(c). In this case, to find k such that 4x + 3 is a factor, we can set x = -3/4 (the root of 4x + 3) and ensure that the polynomial evaluates to zero. This theorem provides a straightforward method to check for factors.

Substitution Method

The substitution method involves replacing a variable in an expression with a specific value to simplify calculations. In this problem, substituting x = -3/4 into the polynomial allows us to create an equation involving k. By solving this equation, we can find the value of k that ensures 4x + 3 is a factor of the given polynomial.

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Related Practice

Textbook Question

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5

Textbook Question

In Exercises 73–74, use the graph of the rational function to solve each inequality.

1/4(x + 2) > - 3/4(x - 2)

Textbook Question

In Exercises 73–74, use the graph of the rational function to solve each inequality.

1/4(x + 2) ≤ - 3/4(x - 2)

Textbook Question

Follow the seven steps to graph each rational function. f(x)=x⁴/(x²+2)

Textbook Question

When 2x²−7x+9 is divided by a polynomial, the quotient is 2x-3 and the remainder is 3. Find the polynomial.

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Textbook Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x²−4)