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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 13
Chapter 4, Problem 13

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)=5x3-3x2+2x-6; k=2

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1
Write down the coefficients of the polynomial ƒ(x) = 5x^3 - 3x^2 + 2x - 6. These are 5, -3, 2, and -6.
Set up the synthetic division by writing the value of k = 2 to the left, and the coefficients in a row to the right: 5, -3, 2, -6.
Bring down the first coefficient (5) as it is. Then multiply this number by k (2) and write the result under the next coefficient: 5 × 2 = 10.
Add the second coefficient (-3) and the number just written (10): -3 + 10 = 7. Repeat the multiply and add process: multiply 7 by 2, write the result under the next coefficient, then add.
Continue this process until all coefficients have been used. The last number you get is the remainder r. The other numbers form the coefficients of the quotient polynomial q(x). Finally, express ƒ(x) as ƒ(x) = (x - 2)q(x) + r.

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Key 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 binomial of the form x - k. It simplifies the long division process by using only the coefficients of the polynomial and performing arithmetic operations in a tabular form. This method quickly yields the quotient and remainder.
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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. The remainder is a constant because the divisor is linear, and this form helps in understanding factorization and roots.
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Evaluating Remainder Using Remainder Theorem

The Remainder Theorem states that the remainder when a polynomial ƒ(x) is divided by (x - k) is equal to ƒ(k). This provides a quick way to find the remainder without completing the entire division, and it confirms the result obtained from synthetic division.
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Related Practice
Textbook Question

Solve each quadratic inequality. Give the solution set in interval notation.

(a) -(x + 1)(x + 2) ≥ 0

(b) -(x + 1)(x + 2) > 0

(c) -(x + 1)(x + 2) ≤ 0

(d) -(x + 1)(x + 2) < 0

Textbook Question

Solve each quadratic inequality. Give the solution set in interval notation. - ( x +√2)(x-3) < 0

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Textbook Question

Consider the graph of each quadratic function.

a) Give the domain and range.

Textbook Question

Use synthetic division to perform each division. (3x3+6x2-8x+3)/(x+3)

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Textbook Question

Solve each problem. Suppose r varies directly as the square of m, and inversely as s. If r=12 when m=6 and s=4, find r when m=6 and s=20.

Textbook Question

Use the graphs of the rational functions in choices A–D to answer each question.

There may be more than one correct choice. If ƒ represents the function, only one choice has a single solution to the equation ƒ(x)=3. Which one is it?