Problem 41 - Algebra 1: Concepts and Skills

Question

Factor the expression completely. \(c^{4}+c^{3}-12 c-12\)

Step-by-Step Solution

Answer

The fully factored form of the polynomial \(c^{4}+c^{3}-12 c-12\) is \((c + 1)(c^3 - 12)\)

Step 1: Combine Like Terms

The given polynomial is \(c^{4}+c^{3}-12 c-12\). First, reorder the expression to combine like terms, this yields : \(c^{4}+c^{3}-12 c-12 = c^4 + c^3 -12c -12\)

Step 2: Group Terms

Next, group the terms as follows: \(c^4 + c^3 -12c -12 = (c^4 + c^3) - (12c + 12)\) by grouping the terms two by two.

Step 3: Factor out Common Terms from Each Binomial

From the first binomial, \(c^4 + c^3\), we can factor a \(c^3\) out. Thus \(c^4 + c^3 = c^3(c + 1)\). From the second binomial \(12c + 12\), by applying the Common Factor Rule in Factoring, we can factor a 12 out, thus \(12c + 12 = 12(c + 1)\). The expression now becomes : \(c^3(c + 1) - 12(c + 1)\)

Step 4: Final Factoring

Our last step would be to factor out the common binomial \((c + 1)\) from the expression \(c^3(c + 1) - 12(c + 1)\), which results in : \((c + 1)(c^3 - 12)\)