Answer:
a)
[tex]-2x^6+7x^4+3x^3-3x^2+11x+20[/tex]b) The product of (-2x^3 + x - 5 ) and (x^3 - 3x - 4 ) is equal to the product of (x^3 - 3x - 4) and (-2x^3 + x - 5 ) since, according to the commutative property of multiplication, changing the order of the numbers being multiplied does not change the product.
Explanation:
Given the expressions;
[tex]\begin{gathered} \mleft(-2x^3+x-5\mright) \\ \text{and } \\ (x^3-3x-4) \end{gathered}[/tex]a) To determine the product of the above expressions, we have to use each term in the first expression to multiply each term in the second expression as seen below below;
[tex]\begin{gathered} (-2x^3+x-5)\times\mleft(x^3-3x-4\mright) \\ =(-2x^3\cdot x^3)+\lbrack-2x^3(-3x)\rbrack+\lbrack-2x^3(-4)\rbrack+(x\cdot x^3)+\lbrack x(-3x)\rbrack \\ +\lbrack x(-4)\rbrack+(-5\cdot x^3)+\lbrack-5(-3x)\rbrack+\lbrack-5(-4)\rbrack \end{gathered}[/tex][tex]\begin{gathered} =-2x^6+6x^4+8x^3+x^4-3x^2-4x-5x^3+15x+20 \\ =-2x^6+7x^4+3x^3-3x^2+11x+20 \end{gathered}[/tex]So the product of (-2x^3 + x - 5 ) and (x^3 - 3x - 4 ) is;
[tex]-2x^6+7x^4+3x^3-3x^2+11x+20[/tex]b) Note that the commutative property of multiplication states that changing the order of the numbers being multiplied does not change the product.
So we can say that the product of (-2x^3 + x - 5 ) and (x^3 - 3x - 4 ) is equal to the product of (x^3 - 3x - 4) and (-2x^3 + x - 5 ) since, according to the commutative property of multiplication, changing the order of the numbers being multiplied does not change the product.
