Question

f(x)=5×3+x2-5x-1,g(x)=x+1​

Answer 1

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Factor Theorem :

Let p(x) be a polynomial if

degree one or more than one

and a is a real number .Then ,

i ) x - a , will be a factor of p(x)

if p(a) = 0 conversely

ii ) If x - a is a factor of p(x),

Then p(a) = 0

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Here ,

F(x) = 5x³ + x² - 5x - 1 ,

g(x) = x + 1 ,

g(x) = 0 ,

=> x + 1 = 0

=> x = -1

Therefore ,

Zero of g(x) is -1 ,

Now ,

F(-1) = 5(-1)³ + (-1)² - 5(-1) - 1

= -5 + 1 + 5 - 1

=-6 + 6

= 0

Therefore ,

By Factor theorem ,

g(x) is a factor of F(x).

•••••

Answer 2

Answer:

Step-by-step explanation:

(f + g)(x) = f (x) + g(x)

= [3x + 2] + [4 – 5x]

= 3x + 2 + 4 – 5x

= 3x – 5x + 2 + 4

= –2x + 6

(f – g)(x) = f (x) – g(x)

= [3x + 2] – [4 – 5x]

= 3x + 2 – 4 + 5x

= 3x + 5x + 2 – 4

= 8x – 2

(f × g)(x) = [f (x)][g(x)]

= (3x + 2)(4 – 5x)

= 12x + 8 – 15x2 – 10x

= –15x2 + 2x + 8

\left(\small{\dfrac{f}{g}}\right)(x) = \small{\dfrac{f(x)}{g(x)}}(

g

f

​

)(x)=

g(x)

f(x)

​

= \small{\dfrac{3x+2}{4-5x}}=

4−5x

3x+2

​