Algebra Q-Test Solutions: Polynomials and Partial Fractions Problems

This document provides solutions to a Q-Test covering key concepts in algebra, specifically focusing on polynomials and partial fractions. The solutions address various problems, including finding the quotient and remainder of a sum, determining if a linear expression is a factor of a polynomial, arranging polynomials in higher order, performing polynomial long division, understanding the applicability of synthetic division, finding partial fractions of given expressions, and identifying conditions for expressing fractional functions in partial fractions. The document also provides detailed explanations and references to support the solutions, making it a valuable resource for students studying algebra and related mathematical topics. It covers topics like the quotient and remainder theorem, factor theorem, polynomial division, and partial fraction decomposition.

Q-TEST
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TABLE OF CONTENTS
1. Quotient and remainder for the sum 115/7..................................................................................2
2. Is (x+2) a factor of the polynomial x^3-3x^2-4x+12..................................................................2
3. Higher order for writing out 4x^2-5x+3x^3-5+5x^4 polynomial................................................2
4. Answer of polynomial long division, 12x^3-11x^2+9x+12 when it is divided by 4x+3............2
5. Is it true or false that synthetic division can always be used when dividing polynomials..........2
6. What are the partial fractions of [x/x-2] + [3/x+1]......................................................................2
7. Before a fractional function can be expressed directly in partial fractions..................................2
8. Is the expression 5x/[(x^2+x+1)(x-2)].........................................................................................2
5x/[(x^2+x+1)(x-2)] is referred as quadratic partial fraction..........................................................2
REFERENCES................................................................................................................................3
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1. QUOTIENT AND REMAINDER FOR THE SUM 115/7
Appropriate evaluation of the sum 115/7 indicates that its quotient and reminder 16 R4.
2. IS (X+2) A FACTOR OF THE POLYNOMIAL X^3-3X^2-4X+12
As per the detailed analysis of polynomial, (x+2) is a factor of polynomial x^3-3x^2-
4x+12. It helps in analysing the final outcome of polynomial.
3. HIGHER ORDER FOR WRITING OUT 4X^2-5X+3X^3-5+5X^4
POLYNOMIAL
5-5x+4x^2-3x^3+5x^4 is considered as a higher order for the writing out 4x^2-5x+3x^3-
5+5x^4 polynomial. Professionals need to ensure about such values in desired manner.
4. ANSWER OF POLYNOMIAL LONG DIVISION, 12X^3-11X^2+9X+12
WHEN IT IS DIVIDED BY 4X+3
According to proper evaluation, it can be state that 3x^2-5x+6 is considered as correct
answer to the polynomial long division, 12x^3-11x^2+9x+12 when it is divided by 4x+3
5. IS IT TRUE OR FALSE THAT SYNTHETIC DIVISION CAN ALWAYS
BE USED WHEN DIVIDING POLYNOMIALS
It is true that synthetic division can always be used when dividing polynomials
6. WHAT ARE THE PARTIAL FRACTIONS OF [X/X-2] + [3/X+1]
Partial fraction for the [x/x-2] + [3/x+1] can be classified as 5x-4/x^2-x-2
7. BEFORE A FRACTIONAL FUNCTION CAN BE EXPRESSED
DIRECTLY IN PARTIAL FRACTIONS
It has been noticed that before a fractional function can be expressed directly in partial
fractions, when the numerator must be greater than the denominator. It allows to attain the
appropriate expression in regard to the services.
8. IS THE EXPRESSION 5X/[(X^2+X+1)(X-2)]
5x/[(x^2+x+1)(x-2)] is referred as quadratic partial fraction
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REFERENCES
Doty, R.L., 2014. University of Pennsylvania Smell Identification Test: a rapid quantitative
olfactory function test for the clinic. The Laryngoscope. 94(2). pp.176-178.
Cieslik, J., 2007. On possibilities of the extension of Kharitonov's stability test for interval
polynomials to the discrete-time case. IEEE Transactions on Automatic Control.32(3).
pp.237-238.
Bartolini, G., 2008. Chattering avoidance by second-order sliding mode control. IEEE
Transactions on Automatic control. 43(2). pp.241-246.
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