Katie's school has a rectangular courtyard whose area can be expressed as 3x2 - 7x + 2. Which of the following could be the dimensions of the courtyard in terms of x?
Since the trinomial does not have a coef?cient of one on its highest exponent term, the simplest way to ?nd out the answer to this problem is to use the distributive property. First, by using the original trinomial, identify the sum and product through looking at the terms when the trinomial is in descending order (highest exponent ?rst): 3x2 - 7x + 2.The sum is the middle term, in this case, 7x.The product is the product of the ?rst and last terms, in this case, (3x2)(2) = 6x2. Now, identify two quantities whose sum is -7x and product is 6x2, namely -6x and -x. Rewrite the original trinomial using these two terms to replace the middle term in any order: 3x2 -6x - x + 2. Now factor by grouping by taking a common factor out of each pair of terms.The common factor of 3x2 and -6x is 3x and the common factor of -x and 2 is -1.Thus, 3x2 - 6x - x + 2 becomes 3x(x - 2) - 1(x - 2). (Remember that if this expression was multiplied back out and simpli?ed, it would correctly yield the original polynomial.)Presently, this two-term expression has a common factor of (x - 2) which can be factored out of each term using the distributive property: 3x(x - 2) - 1(x - 2) becomes (x - 2)(3x -1).The dimensions of the courtyard are (x - 2) and (3x - 1).