Question #13537

1 Answer
May 3, 2017

a) g(5x+2) = 75x^2+35x+4g(5x+2)=75x2+35x+4
b) f(x) - h(5) = 2x-125f(x)h(5)=2x125
c) f(x)[h(x)-g(x)] = 2x^4-11x^3+23x^2-24x+10f(x)[h(x)g(x)]=2x411x3+23x224x+10

Explanation:

Given:

f(x) = 2x-5f(x)=2x5
g(x) = 3x^2-5x+2g(x)=3x25x+2
h(x) = x^3-xh(x)=x3x

a) In order to find g(5x+2)g(5x+2) we must plug in for every value of xx in g(x)g(x) the value (5x+2)(5x+2).

g(x) = 3x^2-5x+2g(x)=3x25x+2
g(5x+2) = 3(5x+2)^2-5(5x+2)+2g(5x+2)=3(5x+2)25(5x+2)+2

Now we simplify the expression on the right hand side.

= 3(5x+2)(5x+2)-5(5x+2)+2=3(5x+2)(5x+2)5(5x+2)+2

We FOIL (color(red)(5x)color(blue)(+2))(color(green)(5x)color(purple)(+2))(5x+2)(5x+2) and distribute (-5)(5) to (5x+2)(5x+2)

= 3[(color(red)(5x))(color(green)(5x))+(color(red)(5x))(color(purple)(+2))+(color(blue)(2))(color(green)(5x))+(color(blue)(+2))(color(purple)(+2))]-25x-10+2=3[(5x)(5x)+(5x)(+2)+(2)(5x)+(+2)(+2)]25x10+2

= 3[25x^2+10x+10x)+4]-25x-8=3[25x2+10x+10x)+4]25x8

= 75x^2+60x+12-25x-8=75x2+60x+1225x8

g(5x+2) = 75x^2+35x+4g(5x+2)=75x2+35x+4

b) For this part, we again substitute the given value of 55 into h(x)h(x)

f(x) = 2x-5f(x)=2x5
h(x) = x^3-xh(x)=x3x

f(x) - h(5) = 2x-5 - [(5)^3-(5)]f(x)h(5)=2x5[(5)3(5)]

= 2x-5 - 125 + 5=2x5125+5

f(x) - h(5) = 2x-125f(x)h(5)=2x125

c) For this part, we need only to perform arithmetic on the given functions:

f(x) = 2x-5f(x)=2x5
g(x) = 3x^2-5x+2g(x)=3x25x+2
h(x) = x^3-xh(x)=x3x

f(x)[h(x)-g(x)] = (2x-5)[x^3-x-(3x^2-5x+2)]f(x)[h(x)g(x)]=(2x5)[x3x(3x25x+2)]

Simplifying the expression in the brackets, we get
= (2x-5)[x^3-x-3x^2+5x-2]=(2x5)[x3x3x2+5x2]
= (2x-5)[x^3-3x^2+4x-2]=(2x5)[x33x2+4x2]

Now we need to expand and multiply (2x-5)(2x5) by [x^3-3x^2+4x-2][x33x2+4x2]

=(color(red)(2x)color(blue)(-5))[x^3-3x^2+4x-2]=(2x5)[x33x2+4x2]

We can distribute the terms in (2x-5)(2x5) and multiply each of them by [x^3-3x^2+4x-2][x33x2+4x2]

=(color(red)(2x))[x^3-3x^2+4x-2]+(color(blue)(-5))[x^3-3x^2+4x-2]=(2x)[x33x2+4x2]+(5)[x33x2+4x2]

This gives:

=color(red)(2x^4-6x^3+8x^2-4x)2x46x3+8x24x
color(blue)(-5x^3+15x^2-20x+10)5x3+15x220x+10

Combining like terms, gives:

=color(red)(2)x^4 + (color(red)(-6)color(blue)(-5))x^3 + (color(red)(8)color(blue)(+15))x^2 + (color(red)(-4)color(blue)(-20))x+color(blue)(10) =2x4+(65)x3+(8+15)x2+(420)x+10

f(x)[h(x)-g(x)] = 2x^4-11x^3+23x^2-24x+10f(x)[h(x)g(x)]=2x411x3+23x224x+10