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Answer :
First, we start with the expression
[tex]$$2x^6 - x^3\left(x^3-4\right).$$[/tex]
Step 1: Expand the expression
Distribute [tex]$x^3$[/tex] inside the parentheses:
[tex]$$2x^6 - \left(x^3 \cdot x^3 - x^3 \cdot 4\right).$$[/tex]
Notice that
[tex]$$x^3 \cdot x^3 = x^6 \quad \text{and} \quad x^3 \cdot 4 = 4x^3.$$[/tex]
So the expression becomes
[tex]$$2x^6 - \left(x^6 - 4x^3\right).$$[/tex]
Now, distribute the negative sign:
[tex]$$2x^6 - x^6 + 4x^3.$$[/tex]
Combine the like terms:
[tex]$$2x^6 - x^6 = x^6,$$[/tex]
thus yielding
[tex]$$x^6 + 4x^3.$$[/tex]
Step 2: Factor the simplified expression
Observe that both terms in the expression have a common factor of [tex]$x^3$[/tex]. Factor [tex]$x^3$[/tex] out:
[tex]$$x^6 + 4x^3 = x^3\left(x^3\right) + x^3\left(4\right) = x^3(x^3 + 4).$$[/tex]
Thus, the expression can also be written as
[tex]$$x^3(x^3 + 4).$$[/tex]
Final Answer:
The expanded form is
[tex]$$x^6 + 4x^3,$$[/tex]
and the fully factored form is
[tex]$$x^3(x^3 + 4).$$[/tex]
[tex]$$2x^6 - x^3\left(x^3-4\right).$$[/tex]
Step 1: Expand the expression
Distribute [tex]$x^3$[/tex] inside the parentheses:
[tex]$$2x^6 - \left(x^3 \cdot x^3 - x^3 \cdot 4\right).$$[/tex]
Notice that
[tex]$$x^3 \cdot x^3 = x^6 \quad \text{and} \quad x^3 \cdot 4 = 4x^3.$$[/tex]
So the expression becomes
[tex]$$2x^6 - \left(x^6 - 4x^3\right).$$[/tex]
Now, distribute the negative sign:
[tex]$$2x^6 - x^6 + 4x^3.$$[/tex]
Combine the like terms:
[tex]$$2x^6 - x^6 = x^6,$$[/tex]
thus yielding
[tex]$$x^6 + 4x^3.$$[/tex]
Step 2: Factor the simplified expression
Observe that both terms in the expression have a common factor of [tex]$x^3$[/tex]. Factor [tex]$x^3$[/tex] out:
[tex]$$x^6 + 4x^3 = x^3\left(x^3\right) + x^3\left(4\right) = x^3(x^3 + 4).$$[/tex]
Thus, the expression can also be written as
[tex]$$x^3(x^3 + 4).$$[/tex]
Final Answer:
The expanded form is
[tex]$$x^6 + 4x^3,$$[/tex]
and the fully factored form is
[tex]$$x^3(x^3 + 4).$$[/tex]
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