Video Transcript
Factor the expression three 𝑝 times 𝑛 cubed plus one minus 𝑛 cubed minus one completely.
To begin, we note that this expression has some partial factorization on the first two terms. Prior to factoring out three 𝑝, the first two terms must have been three 𝑝𝑛 cubed plus three 𝑝. The original polynomial expression has an even number of terms that do not all share a common factor. However, the first two terms share a common factor of three 𝑝. If we can find a common factor of the last two terms, then we may be able to use factoring by grouping to answer this question.
Ideally, we can take out a common factor from the last two terms that leaves us with a factor of 𝑛 cubed plus one in the parentheses. To factor by grouping, we need to identify a common factor in the two pairs of factored terms.
In our first step, we will factor out a negative one from the last two terms. So we show the factorization of the last two terms is negative one times 𝑛 cubed plus one. Then, we can see that 𝑛 cubed plus one is the factor in common to both pairs of factored terms. The common factor of the first two terms and the common factor of the last two terms make up the binomial factor underlined in orange. We have shown the given expression can be factored as three 𝑝 minus one times 𝑛 cubed plus one. However, to make sure we have factored completely, we must check to see if the cubic binomial is prime or factorable.
Since the factor in the second parentheses is of the form 𝑎 cubed plus 𝑏 cubed, we can use the sum of cubes factoring formula, which states that a polynomial of the form 𝑎 cubed plus 𝑏 cubed has factors 𝑎 plus 𝑏 times 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared. In this case, 𝑎 equals 𝑛 and 𝑏 equals one. Following the sum of cubes formula, we have 𝑛 plus one times 𝑛 squared minus 𝑛 times one plus one squared. Simplifying the expression in the second parentheses yields 𝑛 squared minus 𝑛 plus one. Altogether, the complete factorization of the given expression is three 𝑝 minus one times 𝑛 plus one times 𝑛 squared minus 𝑛 plus one. We found our answer through factoring by grouping and using the sum of cubes factoring formula.
Whenever possible, it is a good idea to check our answer by multiplying the factors back together to see if we get the original polynomial expression. We can start our check by multiplying three 𝑝 minus one and 𝑛 plus one, which leads to three 𝑝𝑛 plus three 𝑝 minus 𝑛 minus one. Then, we will carefully multiply this four-term expression with 𝑛 squared minus 𝑛 plus one.
First, distributing three 𝑝𝑛, we get three 𝑝𝑛 cubed minus three 𝑝𝑛 squared plus three 𝑝𝑛. Then, by distributing three 𝑝, we get three 𝑝𝑛 squared minus three 𝑝𝑛 plus three 𝑝. By distributing negative 𝑛, we get negative 𝑛 cubed plus 𝑛 squared minus 𝑛. Then, by distributing negative one, we get negative 𝑛 squared plus 𝑛 minus one. And finally, by canceling like terms with opposite signs, we are left with three 𝑝𝑛 cubed plus three 𝑝 minus 𝑛 cubed minus one. And if we factor three 𝑝 from the first two terms, we have the exact expression given at the start of this problem. This confirms that we have correctly factored the given expression.
