Question

What is the product of 2x^3+x-5 and x^3-3x-4 ?

(a) Show your work.

(b) Is the product of -2x^3+x-5 and x^3-3x-4 equal to the product of x^3-3x-4 and -2x^3+x-5?
 Explain your answer.

Answer 1

FULL ANSWERS WITH STEPS

(a) Product of $2x^{3}+x-5$ and $x^{3}-3x-4$
$(2x^{3}+x-5)(x^{3}-3x-4)$
$=2x^{3}(x^{3}-3x-4)+x(x^{3}-3x-4)-5(x^{3}-3x-4)$
$=2x^{3}*x^{3}+2x^{3}*(-3x)+2x^{3}*(-4)+x*x^{3}+x*(-3x)+x*(-4)-5*x^{3}-5*(-3x)-5*(-4)$
$=2x^{6}-6x^{4}-8x^{3}+x^{4}-3x^{2}-4x-5x^{3}+15x+20$
$=2x^{6}-6x^{4}+x^{4}-8x^{3}-5x^{3}-3x^{2}-4x+15x+20$
$=2x^{6}-5x^{4}-13x^{3}-3x^{2}+11x+20$

(b)
1)Finding the product of $-2x^{3}+x-5$ and $x^{3}-3x-4$
$(-2x^{3}+x-5)(x^{3}-3x-4)$
$=-2x^{3}(x^{3}-3x-4)+x(x^{3}-3x-4)-5(x^{3}-3x-4)$
$=-2x^{3}*x^{3}-2x^{3}*(-3x)-2x^{3}*(-4)+x*x^{3}+x*(-3x)+x*(-4)-5*x^{3}-5*(-3x)-5*(-4)$
$=-2x^{6}+6x^{4}+8x^{3}+x^{4}-3x^{2}-4x-5x^{3}+15x+20$
$=-2x^{6}+6x^{4}+x^{4}+8x^{3}-5x^{3}-3x^{2}-4x+15x+20$
$=-2x^{6}+7x^{4}+3x^{3}-3x^{2}+11x+20$
∴ $(-2x^{3}+x-5)(x^{3}-3x-4)=-2x^{6}+7x^{4}+3x^{3}-3x^{2}+11x+20$  ...(1)

2) Finding the product of $x^{3}-3x-4$ and $-2x^{3}+x-5$
$(x^{3}-3x-4)(-2x^{3}+x-5)$
$=(-2x^{3}+x-5)(x^{3}-3x-4)$
$=-2x^{6}+7x^{4}+3x^{3}-3x^{2}+11x+20$
∴ $(x^{3}-3x-4)(-2x^{3}+x-5)=-2x^{6}+7x^{4}+3x^{3}-3x^{2}+11x+20$  ...(2) [Using (1)]

Now, 
$(-2x^{3}+x-5)(x^{3}-3x-4)=(-2x^{3}+x-5)(x^{3}-3x-4)$
⇒ $(-2x^{3}+x-5)(x^{3}-3x-4)=-2x^{6}+7x^{4}+3x^{3}-3x^{2}+11x+20$  [Using (1)]
⇒ $(-2x^{3}+x-5)(x^{3}-3x-4)=(x^{3}-3x-4)(-2x^{3}+x-5)$    [Using (2)]

∴ Product of $-2x^{3}+x-5$ and $x^{3}-3x-4$ = Product of $x^{3}-3x-4$ and $-2x^{3}+x-5$

Hope these may help you. 

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