Textbook Question
Consider the graph of each quadratic function.
(a) Give the domain and range.
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 14
Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x)=(x-k)q(x)+r. ƒ(x)=-3x3+5x-6; k=-1
Verified step by step guidance1
Write down the coefficients of the polynomial ƒ(x) = -3x^3 + 0x^2 + 5x - 6. Note that the coefficient of x^2 is 0, so the coefficients are: -3, 0, 5, -6.
Set up the synthetic division using k = -1. Write -1 to the left and the coefficients in a row: -3, 0, 5, -6.
Bring down the first coefficient (-3) as it is. Then multiply it by k (-1) and write the result under the next coefficient. Add the column and write the sum below.
Repeat the multiply and add process for each coefficient: multiply the last sum by k (-1), write it under the next coefficient, and add the column.
After completing the synthetic division, the last number you get is the remainder r. The other numbers form the coefficients of the quotient polynomial q(x). Express ƒ(x) as ƒ(x) = (x - k)q(x) + r using these results.
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Video duration:
5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - k. It simplifies the long division process by using only the coefficients of the polynomial and performing arithmetic operations to find the quotient and remainder quickly.
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Polynomial Division Algorithm
The polynomial division algorithm states that for any polynomial ƒ(x) divided by (x - k), there exist a quotient polynomial q(x) and a remainder r such that ƒ(x) = (x - k)q(x) + r. This expresses the original polynomial in terms of its divisor, quotient, and remainder.
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Evaluating Polynomials at a Point
Evaluating a polynomial at x = k helps determine the remainder when dividing by (x - k). According to the Remainder Theorem, the remainder r is equal to ƒ(k), which can be found by substituting k into the polynomial.
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Related Practice
Textbook Question
Use synthetic division to perform each division. (-9x3 + 8x2 - 7x+2) / x-2
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Solve each quadratic inequality. Give the solution set in interval notation. (x-4)(x + √2) < 0
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Use synthetic division to perform each division. (x5 + 3x4 + 2x3 + 2x2 + 3x+1) / x+2
Textbook Question
Match each function with its graph without actually entering it into a calculator. Then, after completing the exercises, check the answers with a calculator. Use the standard viewing window. ƒ(x) = (x - 4)2 - 3
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Textbook Question
Solve each problem. Let a be directly proportional to m and n2, and inversely proportional to y3. If a=9when m=4, n=9, and y=3, find a when m=6, n=2, and y=5.