To match each difference of two cubes expression with its correct factored form, let's first factor each expression:
1. [tex]\(64x^3 - 125\)[/tex]:
- [tex]\(125\)[/tex] is [tex]\(5^3\)[/tex], and [tex]\(64x^3\)[/tex] is [tex]\((4x)^3\)[/tex].
- The factorization of [tex]\(a^3 - b^3\)[/tex] is [tex]\((a - b)(a^2 + ab + b^2)\)[/tex].
- Here, [tex]\(a = 4x\)[/tex] and [tex]\(b = 5\)[/tex].
- So, [tex]\(64x^3 - 125 = (4x - 5)(16x^2 + 20x + 25)\)[/tex].
2. [tex]\(64x^3 - 27\)[/tex]:
- [tex]\(27\)[/tex] is [tex]\(3^3\)[/tex], and [tex]\(64x^3\)[/tex] is [tex]\((4x)^3\)[/tex].
- Using the difference of cubes formula: [tex]\((4x - 3)(16x^2 + 12x + 9)\)[/tex].
3. [tex]\(64x^3 - 1000\)[/tex]:
- [tex]\(1000\)[/tex] is [tex]\(10^3\)[/tex], and [tex]\(64x^3\)[/tex] is [tex]\((4x)^3\)[/tex].
- We factor out common factors: [tex]\(1000 = 2^3 \cdot 5^3\)[/tex] and [tex]\(64 = 8 \cdot 8\)[/tex].
- The difference of cubes factorization is simplified by:
- [tex]\(64x^3 - 1000 = 8 \cdot (8x^3 - 125)\)[/tex].
- So, [tex]\((8)(8x^3 - 125) = 8 \cdot (2x)^3 - 5^3\)[/tex].
- Factor giving: [tex]\(8 (2x - 5) (4x^2 + 10x + 25)\)[/tex].
Now, we match each expression with its factored form:
1. [tex]\(64x^3 - 125\)[/tex] matches with [tex]\((4 x-5)(16 x^2+20 x+25)\)[/tex].
2. [tex]\(64x^3 - 27\)[/tex] matches with [tex]\((4 x-3)(16 x^2+12 x+9)\)[/tex].
3. [tex]\(64x^3 - 1000\)[/tex] matches with [tex]\(8(2 x-5)(4 x^2+10 x+25)\)[/tex].
So, the correct matches are:
[tex]\[
\begin{array}{l l}
64 x^3-125 & \quad (4 x-5)\left(16 x^2+20 x+25\right) \\
64 x^3-27 & \quad (4 x-3)\left(16 x^2+12 x+9\right) \\
64 x^3-1,000 & \quad 8(2 x-5)\left(4 x^2+10 x+25\right) \\
\end{array}
\][/tex]
