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Consider the division of two polynomials: f(x) + (x- c). The result of the synthetic division process is shown here. Write the polynomials representing the (a) Dividend, (b) Divisor, (c) Quotient, and (d) Remainder. 5 1 -6 3 4 32 5 -5 - 10 -30 1 -1 -2 -6 2

Consider the division of two polynomials: f(x) + (x- c). The result of the synthetic division process is shown here. Write the polynomials representing the (a) Dividend, (b) Divisor, (c) Quotient, and (d) Remainder. 5 1 -6 3 4 32 5 -5 - 10 -30 1 -1 -2 -6 2

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Consider the division of two polynomials: \( f(x) \div (x - c) \). The result of the synthetic division process is shown here. Write the polynomials representing the (a) Dividend, (b) Divisor, (c) Quotient, and (d) Remainder. The following synthetic division setup shows the steps taken: ``` 5 | 1 -6 3 4 32 | 5 -5 -10 -30 ------------------- 1 -1 -2 -6 2 ``` ### Explanation: 1. **Dividend**: This is the polynomial \( f(x) \) that is being divided. In this case, we start with the polynomial \( f(x) = x^4 - 6x^3 + 3x^2 + 4x + 32 \). 2. **Divisor**: The divisor for synthetic division is in the form \( (x - c) \). Here, \( c \) is given as \( 5 \), so the divisor is \( (x - 5) \). 3. **Quotient**: This is the result of the division process excluding the remainder. From the final row of the synthetic division, the coefficients represent the quotient polynomial: \( x^3 - x^2 - 2x - 6 \). 4. **Remainder**: The remaining value after the division is complete. In this case, the remainder is \( 2 \). Thus, the polynomials are: - (a) **Dividend**: \( f(x) = x^4 - 6x^3 + 3x^2 + 4x + 32 \) - (b) **Divisor**: \( x - 5 \) - (c) **Quotient**: \( x^3 - x^2 - 2x - 6 \) - (d) **Remainder**: \( 2 \)
Transcribed Image Text:Consider the division of two polynomials: \( f(x) \div (x - c) \). The result of the synthetic division process is shown here. Write the polynomials representing the (a) Dividend, (b) Divisor, (c) Quotient, and (d) Remainder. The following synthetic division setup shows the steps taken: ``` 5 | 1 -6 3 4 32 | 5 -5 -10 -30 ------------------- 1 -1 -2 -6 2 ``` ### Explanation: 1. **Dividend**: This is the polynomial \( f(x) \) that is being divided. In this case, we start with the polynomial \( f(x) = x^4 - 6x^3 + 3x^2 + 4x + 32 \). 2. **Divisor**: The divisor for synthetic division is in the form \( (x - c) \). Here, \( c \) is given as \( 5 \), so the divisor is \( (x - 5) \). 3. **Quotient**: This is the result of the division process excluding the remainder. From the final row of the synthetic division, the coefficients represent the quotient polynomial: \( x^3 - x^2 - 2x - 6 \). 4. **Remainder**: The remaining value after the division is complete. In this case, the remainder is \( 2 \). Thus, the polynomials are: - (a) **Dividend**: \( f(x) = x^4 - 6x^3 + 3x^2 + 4x + 32 \) - (b) **Divisor**: \( x - 5 \) - (c) **Quotient**: \( x^3 - x^2 - 2x - 6 \) - (d) **Remainder**: \( 2 \)
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