Question 409239
{{{(u^2+v-w)(u^2-v+w)}}}
The difference of squares pattern is:
{{{(a+b)(a-b) = a^2 - b^2}}}
To use this pattern on your expression, we need to be able to rewrite your expression (at least in our heads) as the sum of an "a" and a "b" times the difference of that "a" and "b". Once you realize that the "a" and "b" can be <i>any</i> Math expression, then you will become a more powerful user of patterns.<br>
Here's how we can rewrite your expression:
{{{(u^2+(v-w))(u^2-(v-w))}}}
Take a moment to see how this expression is equal to your original expression. Note how the minus in front of the parentheses in the second factor make the "-w" inside the parentheses equal to the "+w" in your original expression.<br>
Once written this way, it is not hard to see that we have matched the difference of squares pattern (the left side) with the "a" being {{{u^2}}} and the "b" being (v-w). So we can use the pattern to multiply, knowing that the answer will be difference of the squares of the "a" and the "b":
{{{(u^2)^2 - (v-w)^2}}}
Squaring {{{u^2}}} is simple. To square (v-w) we can use another pattern: {{{(a-b)^2 = a^2 -2vw + w^2}}}:
{{{u^4 - (v^2 -2vw + w^2)}}}
Note the use of parentheses. That whole entire expression is {{{(v-w)^2}}}. And if we were subtracting {{{(v-w)^2}}} before then we need to subtract the whole expression when we replace {{{(v-w)^2}}}.
One last simplification:
{{{u^4 - v^2 + 2vw - w^2)}}}<br>
The alternative to using the pattern to multiply is to multiply the trinomials the "normal" way: Multiply each term of one polynomial by each term of the other and then add like terms,if any. This would mean 9 multiplications plus adding like terms. Using the patterns, once you learn how, makes this much easier.