Question 203108
{{{4x^3-28x^2+49x}}}
Factor out the GCF:
{{{4x^3 = 2*2*x*x*x}}}
{{{28x^2 = 2*2*7*x*x}}}
{{{49x = 7*7*x}}}
The only factor common to all three terms is x. So the GCF is x. Factoring this out we get:
{{{4x^3-28x^2+49x = x(4x^2 -28x + 49)}}}<br>
Next we will try to factor {{{4x^2 -28x + 49}}} using patterns. The patterns which are most often taught are:<ul><li>Difference of squares: {{{a^2 - b^2 = (a + b)(a - b)}}}</li><li>Difference of cubes: {{{a^3 - b^3 = (a - b)(a^2 + ab + b^2)}}}</li><li>Sum of cubes: {{{a^3 + b^3 = (a + b)(a^2 - ab + b^2)}}}</li><li>Perfect square trinomials:<ul><li>{{{a^2 + 2ab + b^2 = (a + b)(a + b) = (a + b)^2}}}</li><li>{{{a^2 - 2ab + b^2 = (a - b)(a - b) = (a - b)^2}}}</li></ul></li></ul>
Since the first three patterns factor binomials (two-term expressions) and since we have a trinomial (three-term expression), we will not be able to use the first three patterns. And since the fourth pattern has a "+" in front of the middle term and we have a "-", that pattern is out, too. The only possible pattern for {{{4x^2 -28x + 49}}} is the last one: {{{a^2 -2ab + b^2 = (a - b)^2}}}.<br>
To see if our expression fits the pattern:<ol><li>The first term in the pattern, {{{a^2}}} means that the first term in our trinomial must be a perfect square. Since {{{4x^2 = (2x)^2}}} it is a perfect square. It is "2x squared". So our "a" is 2x.</li><li>The last term in the pattern is {{{b^2}}}, a perfect square. So our last term has to be a perfect square (or we will not be able to use this pattern). Our last term, 49, is 7 squared. So it is a perfect square and our "b" is 7.</li><li>Last of all we need to see if our middle term fits the pattern's middle term which is 2ab. Since our "a" is 2x and our "b" is 7, 2ab = 2*(2x)*7 = 28x, our middle term!</li></ol>
Since all three terms of our trinomial fit the pattern of {{{a^2 -2ab + b^2}}} with "a" = 2x and "b" = 7, our trinomial <b>will</b> factor according to this pattern:
{{{a^2 - 2ab + b^2 = (a - b)(a - b) = (a - b)^2}}}
{{{4x^2 -28x + 49 = (2x)^2 -2(2x)(7) + 7^2 = (2x - 7)^2}}}
Now our complete expression is:
{{{x(4x^2 -28x + 49) = x(2x - 7)^2}}}
Remember, factoring is like reducing fractions: You keep going it until you can't go it any further. However, since {{{x(2x - 7)^2}}} will not factor any further, we are finished.