Question

Sums and differences of cubes can be factored using the following patterns. Sum of cubes pattern: \(a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\) Difference of cubes pattern: \(a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\) Use the patterns above to factor the cubic expression completely. Use the distributive property to verify your results. $$ 216-343 t^{3} $$

Short answer

The factors of the expression \(216 - 343t^{3}\) are \((6 - 7t)(36 + 42t + 49t^2)\)

Step-by-step solution

Indentify the Cubes

In the given expression, we identify the cubes. The cube roots of 216 and 343 are 6 and 7 respectively. The cube root of \(t^{3}\) is t. So, the expression \(216 - 343t^{3}\) can be rewritten as \((6)^3 - (7t)^3\).

Apply Difference of Cubes Formula

Next, we apply the difference of cubes formula \(a^{3} - b^{3} = (a - b)(a^2 + ab + b^2)\). By applying this formula to the expression, when a = 6 and b = 7t, we get \((6 - 7t)((6)^{2} + 6*7t + (7t)^{2})\)

Simplify Expression

Finally, we simplify the obtained equation. Squaring 6, and \(7t\), and multiplying 6 with 7t we get \((6 - 7t)(36 + 42t + 49t^2)\) which is the expanded form of the given expression.

Verify Result by Using the Distributive Property

We can verify the results by applying the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. Multiply (6 - 7t) with each term inside the brackets separately. So, \((6 - 7t)(36) + (6 - 7t)(42t) + (6 - 7t)(49t^2) = 216 - 252t + 252t - 343t^3 = 216 - 343t^3\), which matches the original expression.