Example 1

Factor «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#191919¨»«mi mathcolor=¨#191919¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#191919¨»-«/mo»«mi mathcolor=¨#191919¨»x«/mi»«mo mathcolor=¨#191919¨»-«/mo»«mn mathcolor=¨#191919¨»6«/mn»«/math».

The «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»b«/mi»«/math»-value is −1 and the «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»c«/mi»«/math»-value is −6. Find «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»p«/mi»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»q«/mi»«/math» such that «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»p«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mi mathcolor=¨#B94A48¨»q«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»p«/mi»«mi mathcolor=¨#B94A48¨»q«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»6«/mn»«/math» in order to match the previous pattern. Sometimes these values can be determined quickly by inspection. Sometimes a table is more helpful. Start with «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»p«/mi»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»q«/mi»«/math» values that multiply to give −6 and then check to see which pairing gives «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»p«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mi mathcolor=¨#B94A48¨»q«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«/math».

In the last line of the table, «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»p«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mi mathcolor=¨#B94A48¨»q«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»p«/mi»«mi mathcolor=¨#B94A48¨»q«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»6«/mn»«/math». Now, a «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»p«/mi»«/math»-value and a «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»q«/mi»«/math»-value that satisfy the requirements of factoring the trinomial  «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»-«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»6«/mn»«/math» are known. Use these to write the trinomial in factored form.

Verify by expanding.



Example 2

Factor the trinomial .

The b-value is 1 and the c-value is 1. Look for p and q such that pq = 1 and p + q = 1.

This unit focuses on integer values for p and q. However, some trinomials can be factored using non-integer values for p and q.There are no other integer values of p and q that have a product of 1. The trinomial cannot be factored in the form using integer values of p and q .