Question

verify the division algorithm for polynomials p of X equal to 2 x power 4 - 6 x cube + 2 x square minus x + 2 and g( X )=x+ 2​

Answer 1

GIVEN :

The polynomials $p(x)=2x^4-6x^3+2x^2-x+2$ and $g(x)=x+2$

TO VERIFY :

The Division Algorithm for the given polynomials.

SOLUTION :

Given that the polynomials $p(x)=2x^4-6x^3+2x^2-x+2$ and $g(x)=x+2$

Now divide the polynomial p(x) by g(x)

          $2x^3-10x^2+22x-45$

      _________________________

x+2 ) $2x^4-6x^3+2x^2-x+2$

        $2x^4+4x^3$

      _(-)___(-)______________

                  $-10x^3+2x^2$

                  $-10x^3-20x^2$

              __(+)___(+)_________________

                                 $22x^2-x$

                                 $22x^2+44x$

                          ___(-)___(-)________

                                            $-45x+2$

                                            $-45x-90$

                                       ___(+)__(+)_________

                                                         92

                                              _______________

∴ quotient is $2x^3-10x^2+22x-45$ and remainder =92

Now we can verify the Division Algorithm:

$Dividend=quotient\times divisor+remainder$

Substitute the values in the formula we get

$2x^4-6x^3+2x^2-x+2=(2x^3-10x^2+22x-45)\times (x+2)+92$

By using the Distributive property :

(x+y+z)(a+b)=(x+y+z)(a)+(x+y+z)(b)

$2x^4-6x^3+2x^2-x+2=(2x^3-10x^2+22x-45)(x)+(2x^3-10x^2+22x-45)(2)+92$

$=2x^3(x)-10x^2(x)+22x(x)-45(x)+2x^3(2)-10x^2(x)+22x(2)-45(2)+92$

$=2x^4-10x^3+22x^2-45x+4x^3-10x^2+44x-90+92$

Adding the like terms

$2x^4-6x^3+2x^2-x+2=2x^4-6x^3+2x^2-x+2$

Hence LHS = RHS

The division Algorithm is verified.

Answer 2

Step-by-step explanation:

2x²³ - 10x² + 22x - 451 x + 2) 2x² - 6x² + 2x²-x+ 2 -2x4 + 4x³ 3 -10x²³ + 2x²

10x³ - 202²

22x²-x

- 22x²+44x

-45x+2

-45x-90 92

2x² - 6x² +22²-2+2 = (x+2)x(22³-102² +224-45) +92 = 2x² - 6x²³² + 2x²-x+ 2 = 2x²-102²³ +222²-45x + 4x³ - 20x² +44x-90+92

2x² - 6x³ +22²-x+2 = 2x²4 - 10x²³² +222²-45x+4x²³ - 20x² + 44x + 2 -6x² + 2x² - x + 2 = -10x²³ +22x² - 45x + 4x³ - 20x² + 44 x + 2

-6x²³²³ + 2x²_ x = - 6x³ + 2x²-x Hence verified