Question 711099
{{{(2x+5)^3}}}
There are several ways to find this product:<ul><li>Rewrite it as (2x+5)(2x+5)(2x+5) and then multiply it out by hand. (The first multiplication can be done with FOIL (or the {{{(a+b)^2 = a^2+2ab+b^2}}} pattern). But when you multiply that by the third (2x+5) you will have to use the general way to multiply polynomials: Multiply each term of one factor by each term of the other factor and then add like terms, if any.)</li><li>Use the Binomial Theorem, if you know it, which provides a formula for {{{(a+b)^n}}}. In this case we would use 3 for the "n".</li><li>Most students are expected to learn the binomial square pattern: {{{(a+b)^2 = a^2+2ab+b^2}}}. Fewer students go beyond this to memorize the pattern for the cube of a binomial. If you happen to know this pattern, this would be the easiest way to get to the answer.</li></ul>I'm going to assume that you do not know the last two methods (which are shorter than the first). So I will multiply this out by hand.<br>
{{{(2x+5)(2x+5)(2x+5)}}}
I will, however, take advantage of the binomial square pattern to do the first multiplication:
{{{((2x)^2 +2(2x)(5)+(5)^2)(2x+5)}}}
which simplifies to:
{{{(4x^2 +20x+25)(2x+5)}}}<br>
Now we do the second multiplication:
{{{4x^2*2x+4x^2*5+20x*2x+20x*5+25*2x+25*5}}}
which simplifies as follows:
{{{8x^3+20x^2+40x^2+100x+50x+125}}}
{{{8x^3+60x^2+150x+125}}}<br>
P.S. If you're curious, the pattern for a binomial cube is:
{{{(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}}}
Using this pattern on your expression, with the "a" being "2x" and the "b" being "5", we get:
{{{(2x)^3 + 3(2x)^2(5) + 3(2x)(5)^2 + (5)^3}}}
Simplifying:
{{{8x^3 + 3(4x^2)(5) + 3(2x)(25) + 125}}}
{{{8x^3 + 60x^2 + 150x + 125}}}