Answer :
The polynomial [tex]\( x^3 + 64 \)[/tex] is an example of a sum of cubes. Here's how you can understand and factor it:
1. Identify the Form: The sum of cubes is given by the formula [tex]\( a^3 + b^3 \)[/tex]. In this problem, we recognize [tex]\( x^3 + 64 \)[/tex] as a sum of cubes where [tex]\( a = x \)[/tex] and [tex]\( b = 4 \)[/tex], since [tex]\( 64 = 4^3 \)[/tex].
2. Use the Sum of Cubes Formula: The formula for factoring a sum of cubes is:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
In this case:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 4 \)[/tex]
3. Substitute into the Formula:
- [tex]\( a + b = x + 4 \)[/tex]
- [tex]\( a^2 = x^2 \)[/tex]
- [tex]\( ab = x \cdot 4 = 4x \)[/tex]
- [tex]\( b^2 = 4^2 = 16 \)[/tex]
Substitute these values into the formula:
[tex]\[
(x + 4)(x^2 - 4x + 16)
\][/tex]
4. Final Factored Form: Therefore, the polynomial [tex]\( x^3 + 64 \)[/tex] can be factored as:
[tex]\[
(x + 4)(x^2 - 4x + 16)
\][/tex]
This step-by-step process helps in understanding how the polynomial [tex]\( x^3 + 64 \)[/tex] is a sum of cubes and how it is factored using the sum of cubes formula.
1. Identify the Form: The sum of cubes is given by the formula [tex]\( a^3 + b^3 \)[/tex]. In this problem, we recognize [tex]\( x^3 + 64 \)[/tex] as a sum of cubes where [tex]\( a = x \)[/tex] and [tex]\( b = 4 \)[/tex], since [tex]\( 64 = 4^3 \)[/tex].
2. Use the Sum of Cubes Formula: The formula for factoring a sum of cubes is:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
In this case:
- [tex]\( a = x \)[/tex]
- [tex]\( b = 4 \)[/tex]
3. Substitute into the Formula:
- [tex]\( a + b = x + 4 \)[/tex]
- [tex]\( a^2 = x^2 \)[/tex]
- [tex]\( ab = x \cdot 4 = 4x \)[/tex]
- [tex]\( b^2 = 4^2 = 16 \)[/tex]
Substitute these values into the formula:
[tex]\[
(x + 4)(x^2 - 4x + 16)
\][/tex]
4. Final Factored Form: Therefore, the polynomial [tex]\( x^3 + 64 \)[/tex] can be factored as:
[tex]\[
(x + 4)(x^2 - 4x + 16)
\][/tex]
This step-by-step process helps in understanding how the polynomial [tex]\( x^3 + 64 \)[/tex] is a sum of cubes and how it is factored using the sum of cubes formula.
