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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 24
Chapter 4, Problem 24

Divide using synthetic division. (x5+4x4−3x2+2x+3)÷(x−3)

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Identify the divisor and rewrite it in the form \(x - c\). Here, the divisor is \(x - 3\), so \(c = 3\).
Write down the coefficients of the dividend polynomial \(x^{5} + 4x^{4} + 0x^{3} - 3x^{2} + 2x + 3\). Note that the \(x^{3}\) term is missing, so its coefficient is 0. The coefficients are: \([1, 4, 0, -3, 2, 3]\).
Set up the synthetic division by writing \(c = 3\) to the left and the coefficients in a row to the right.
Bring down the first coefficient (1) as it is. Then multiply it by \(c\) (3) and write the result under the next coefficient. Add the column and write the sum below. Repeat this multiply-and-add process for all coefficients.
After completing the process, the last number you get is the remainder. The other numbers form the coefficients of the quotient polynomial, which will be of degree one less than the original polynomial (degree 4 in this case).

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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 - c). It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone.
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Polynomial Coefficients and Missing Terms

When performing synthetic division, it is important to include coefficients for all powers of x, even if some terms are missing. For example, if the polynomial lacks an x³ term, its coefficient is zero, which must be included to maintain proper alignment during division.
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Remainder and Quotient Interpretation

The result of synthetic division provides a quotient polynomial and a remainder. The quotient has one degree less than the original polynomial, and the remainder is a constant. Understanding how to interpret these results is essential for solving division problems and further factorization.
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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. −x2 + 2x ≥ 0

Textbook Question

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; 1 and 5i are zeros; f(-1) = -104

Textbook Question

In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=−11x4−6x2+x+3f(x)=−11x^4−6x^2+x+3

Textbook Question

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=2(x−5)(x+4)2

Textbook Question

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=x/x(x+4)

Textbook Question

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x4−2x3−5x2+8x+4=0

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