Question

On dividing the polynomial 4x4 - 3x3 - 42x2 - 55x -17 by the polynomial g(x) the quotient is x2 - 3x -5 and the remainder is 5x+8. Find g(x)​

Answer 1

$g(x) = 4 {x}^{2} + 9x + 5$

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Answer 2

Answer:

The polynomial g(x) = 4x² + 9x+5

Step-by-step explanation:

Given,

When the polynomial 4x⁴- 3x³ - 42x² - 55x -17 is divided by g(x) the quotient obtained is   x² - 3x -5 and the remainder 5x+8

To find,

The value of g(x)

Solution

Recall the concept

Division lemma

When a polynomial p(x) is divided by another polynomial g(x), then there exist two polynomials q(x) and r(x) such that

p(x) = g(x)q(x) +r(x) --------------(1), the degree of the polynomial r(x) is less than g(x)

Here,

p(x) = 4x⁴- 3x³ - 42x² - 55x -17

q(x) = x² - 3x -5

r(x) =  5x+8

Substituting in equation(1) we get

4x⁴- 3x³ - 42x² - 55x -17  =( x² - 3x -5 )g(x) + 5x+8

( x² - 3x -5 )g(x) = 4x⁴- 3x³ - 42x² - 55x -17 -5x -8

( x² - 3x -5 )g(x) = 4x⁴- 3x³ - 42x² - 60x -25

g(x) = $\frac{4x^4- 3x^3 - 42x^2 - 60x -25}{( x^2 - 3x -5 )}$

Dividing 4x⁴- 3x³ - 42x² - 60x -25 by x² - 3x -5 by long division, we get                  

                 

                                  4x² + 9x+5                  

                x² - 3x -5)   4x⁴- 3x³ - 42x² - 60x -25

                                    4x⁴- 12x³ -20x²

                                   ______________________

                                           9x³ - 22x² - 60x

                                           9x³ - 27x² - 45x      

                                           ______________________                    

                                                  +5x² - 15x - 25

                                                   5x² - 15x - 25

                                                   _________________________  

                                                                      0

g(x) = $\frac{4x^4- 3x^3 - 42x^2 - 60x -25}{( x^2 - 3x -5 )}$ = 4x² + 9x+5

The polynomial g(x) = 4x² + 9x+5

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