Question

f ( x ) = 2 x³ +6 x² + 4 x

g ( x ) = x + 3 x + 2

The polynomials f ( x ) and g ( x ) are defined above. Which of the following polynomials is divisible by 2 x + 3 ?

1). h ( x ) = f ( x ) + g ( x )

2). p ( x ) = f ( x ) + 3 g ( x )

3). r ( x ) = 2 f ( x ) + 3 g ( x )

4). s ( x ) = 3 f ( x ) + 2 g ( x )

Answer 1

$\:\:\:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \huge\mathcal{\bf{{\underline{\underline{\huge\mathcal{...Answer...}}}}}}$

now .. applying remainder theorem....

2x+3=0

X==-3/2

now the remainder when f(X) is divided by 2x+3 is

=2(-3/2)³+6(-3/2)²+4(-3/2)

=(-27/4)+54/4-12/4

={-9+54-12}/4

=-15/4≠0

now the remainder when g(X) is divided by 2x+3 is

=(-3/2)²+3(-3/2)+2

=(9/4)-(9/2)+2

=(9-18+4)/4

=-5/4≠0

so both are not the factor of 2x+3

now .....

1). h ( x ) = f ( x ) + g ( x )

=2 x³ +6 x² + 4 x +x ²+ 3 x + 2

=2x³+7x²+7x+2

therefore..

h(-3/2)=2(-3/2)³+7(-3/2)²+7(-3/2)+2

=(-54/4)+(63/4)-(21/2)+2

=[-54+63-42+8]/4≠0

2). p ( x ) = f ( x ) + 3 g ( x )

=2 x³ +6 x² + 4 x +3(x²+ 3 x + 2)

=2x³+9x²+13x+6

therefore

p(-3/2)=2(-3/2)³+9(-3/2)²+13(-3/2)+6=0

p(-3/2)=0

therefore 2x+3 is a factor of P(X)

Answer:=option(2). p ( x ) = f ( x ) + 3 g ( x )

$\underline{\large\mathcal\red{solution}}$

Answer 2

$\color{Blue}{A.T.Q}$

$\boxed{2x+3=0}$

$\boxed{x=-3/2}$

now the remainder when f(X) is divided by 2x+3 is

2(-3/2)³+6(-3/2)²+4(-3/2)

(-27/4)+54/4-12/4

{-9+54-12}/4

-15/4 is not equal to (0)

now the remainder when g(X) is divided by 2x+3 is

(-3/2)²+3(-3/2)+2

(9/4)-(9/2)+2

(9-18+4)/4

-5/4 is not equal to (0)

so both are not the factor of 2x+3

now .

1). h ( x ) = f ( x ) + g ( x )

=2 x³ +6 x² + 4 x +x ²+ 3 x + 2

=2x³+7x²+7x+2

$\underline {Now}$

h(-3/2)=2(-3/2)³+7(-3/2)²+7(-3/2)+2

(-54/4)+(63/4)-(21/2)+2

[-54+63-42+8]/4 is not equal to (0)

$\boxed { 2nd \: part}$

2) p ( x ) = f ( x ) + 3 g ( x )

2 x³ +6 x² + 4 x +3(x²+ 3 x + 2)

2x³+9x²+13x+6

$\underline {Now}$

p×-3/2)=2

(-3/2)³+9(-3/2)²+13(-3/2)+6=0

p×-3/2)=0

therefore 2x+3 is a factor of value P

Answer:=option(2).

$\huge{P( X) =F(X)+3G( X )}$

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