Verified Answer :

  • Factor out the greatest common divisor (6) from the expression to obtain \(6(36p^2 - 49)\).
  • Recognize that \(36p^2 - 49\) is a difference of squares of the form \((6p)^2 - 7^2\).
  • Apply the difference of squares identity \((a^2 - b^2) = (a - b)(a + b)\) to factor the binomial.
  • Write the final fully factored expression as \(\boxed{6(6p-7)(6p+7)}\).

Explanation

  1. Identifying Common Factors

We are asked to factor the expression \(216p^2 - 294\) completely. Our first step is to identify any common factors shared by the numerical coefficients, which are 216 and 294. Finding the greatest common divisor (GCD) helps simplify the expression significantly before we look for more advanced patterns.

  1. Factoring out the GCD

By finding the prime factorization or systematically dividing, we determine that the greatest common divisor of 216 and 294 is 6. Dividing both terms by 6, we get:
\[216p^2 - 294 = 6(36p^2 - 49)\]
Now the expression inside the parentheses is \(36p^2 - 49\). We can see that both 36 and 49 are perfect squares, as \(36 = 6^2\) and \(49 = 7^2\). This means the expression inside the parentheses takes the form of a difference of squares: \(a^2 - b^2\).

  1. Applying the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). In our case, \(a^2 = 36p^2 = (6p)^2\) and \(b^2 = 49 = 7^2\). Therefore, we can rewrite \(36p^2 - 49\) as \((6p)^2 - 7^2\), which factors into:
\[(6p - 7)(6p + 7)\]
Combining this with the factor of 6 we set aside earlier, the completely factored form is:
\[6(6p - 7)(6p + 7)\]

  1. Conclusion and Verification

We have factored the expression \(216p^2 - 294\) completely. The result of our factorization is \(6(6p - 7)(6p + 7)\). Multiplying these back together gives \(6(36p^2 - 49) = 216p^2 - 294\), confirming our result. The final answer is \(\boxed{6(6p-7)(6p+7)}\).

Examples

Factoring expressions like \(216p^2 - 294\) is a foundational skill in algebra, useful for simplifying complex equations in physics and engineering. For example, when calculating the kinetic energy of an object at different points in a trajectory or determining the cross-sectional area of components in structural design, you often need to simplify polynomial expressions to isolate variables or solve for unknown dimensions. By breaking down large expressions into simpler factors, you gain a better understanding of the underlying relationships between variables.