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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 1a
Chapter 4, Problem 1a

Divide using long division. State the quotient, and the remainder, r(x). (x2+8x+15)÷(x+5)

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Step 1: Set up the long division. Write the dividend \(x^2 + 8x + 15\) under the long division symbol and the divisor \(x + 5\) outside the symbol.
Step 2: Divide the first term of the dividend \(x^2\) by the first term of the divisor \(x\). This gives \(x\). Write \(x\) as the first term of the quotient.
Step 3: Multiply \(x\) (the first term of the quotient) by the divisor \(x + 5\). This gives \(x^2 + 5x\). Subtract \(x^2 + 5x\) from \(x^2 + 8x + 15\), which results in \(3x + 15\).
Step 4: Divide the first term of the new dividend \(3x\) by the first term of the divisor \(x\). This gives \(3\). Write \(3\) as the next term of the quotient.
Step 5: Multiply \(3\) (the next term of the quotient) by the divisor \(x + 5\). This gives \(3x + 15\). Subtract \(3x + 15\) from \(3x + 15\), which results in \(0\). The quotient is \(x + 3\) and the remainder \(r(x)\) is \(0\).

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Key Concepts

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

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. It involves a process similar to numerical long division, where you divide the leading term of the dividend by the leading term of the divisor, multiply the entire divisor by this result, and subtract it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.
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Quotient and Remainder

In polynomial division, the quotient is the result of the division, while the remainder is what is left over after the division process. The relationship can be expressed as: Dividend = Divisor × Quotient + Remainder. Understanding how to identify and express both the quotient and the remainder is crucial for accurately completing polynomial long division.
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Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, which can simplify division problems. In the given exercise, recognizing that the polynomial x^2 + 8x + 15 can be factored into (x + 3)(x + 5) can provide insight into the division process. This understanding can also help verify the results of the long division by checking if the product of the quotient and divisor plus the remainder equals the original polynomial.
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Related Practice
Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x−4)(x+2)>0

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In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = (x + 4)^2 - 2

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Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)(x−5)>0

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Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies inversely as x. y = 12 when x = 5. Find y when x = 2.

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