Question 629303
your expression is:
(x-1/2)(4x^3-2x^2+5x-6) 
you use the distributive law of multiplication.
that law is:
(a + b ) * (c + d + e) equals:
a * (c + d + e) plus:
b * (c + d + e)
you use the distributive law of multiplication again to get:
a * (c + d + e) equals:
a*c + a*d + a*e
and:
b * (c + d + e) equals:
b*c + b*d + b*e.
put them together and you get:
(a + b) * (c + d + e) equals:
a*c + a*d + a*e + b*c + b*d + b*e
we can apply this to your expression of:
(x-1/2)(4x^3-2x^2+5x-6)
using the distributive law of multiplication, this becomes:
x*(4x^3-2x^2+5x-6) - (1/2)(4x^3-2x^2+5x-6)
using the distributive law of multiplication again, this becomes:
x*4x^3 - x*2x^2 + x*5x - x*6 - (1/2)*4x^3 + (1/2)*2x^2 - (1/2)*5x + (1/2)*6
simplifying each term gets you:
4x^4 - 2x^3 + 5x^2 - 6x -2x^3 + x^2 - (5/2)x + 3
you now want to combine like terms if there are any.
-2x^3 and -2x^3 combine into -4x^3
5x^2 and x^2 combine into 6x^2
-6x and -(5/2)x combine into (-17/2)x
putting all the combined terms together, you get:
4x^4 - 4x^3 + 6x^2 - (17/2)x + 3
that's your final expression.
i was able to confirm graphically that this expression and the starting expression are the same.
here's a graph of your starting expression of:
(x-1/2)(4x^3-2x^2+5x-6) 
{{{graph(600,600,-10,10,-50,50,(x-1/2)(4x^3-2x^2+5x-6))}}}
here's a graph of your final expression of:
4x^4 - 4x^3 + 6x^2 - (17/2)x + 3
{{{graph(600,600,-10,10,-50,50,4x^4 - 4x^3 + 6x^2 - (17/2)x + 3)}}}
here's both expressions superimposed on the same graph:
{{{graph(600,600,-10,10,-50,50,4x^4 - 4x^3 + 6x^2 - (17/2)x + 3,(x-1/2)(4x^3-2x^2+5x-6))}}}
since they superimpose on each other perfectly so that they look like one expression, they are identical to each other.
this confirms the solution is correct.