Question 626189


Looking at the expression {{{b^4+2a^2b^2+a^4}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{2}}}, and the last coefficient is {{{1}}}.



Now multiply the first coefficient {{{1}}} by the last coefficient {{{1}}} to get {{{(1)(1)=1}}}.



Now the question is: what two whole numbers multiply to {{{1}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{2}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{1}}} (the previous product).



Factors of {{{1}}}:

1

-1



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{1}}}.

1*1 = 1
(-1)*(-1) = 1


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{2}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=red>1</font></td><td  align="center"><font color=red>1</font></td><td  align="center"><font color=red>1+1=2</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-1+(-1)=-2</font></td></tr></table>



From the table, we can see that the two numbers {{{1}}} and {{{1}}} add to {{{2}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{1}}} both multiply to {{{1}}} <font size=4><b>and</b></font> add to {{{2}}}



Now replace the middle term {{{2a^2b^2}}} with {{{a^2b^2+a^2b^2}}}. Remember, {{{1}}} and {{{1}}} add to {{{2}}}. So this shows us that {{{a^2b^2+a^2b^2=2a^2b^2}}}.



{{{b^4+highlight(a^2b^2+a^2b^2)+a^4}}} Replace the second term {{{2a^2b^2}}} with {{{a^2b^2+a^2b^2}}}.



{{{(b^4+a^2b^2)+(a^2b^2+a^4)}}} Group the terms into two pairs.



{{{b^2(b^2+a^2)+(a^2b^2+a^4)}}} Factor out the GCF {{{b^2}}} from the first group.



{{{b^2(b^2+a^2)+a^2(b^2+a^2)}}} Factor out {{{a^2}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(b^2+a^2)(b^2+a^2)}}} Combine like terms. Or factor out the common term {{{b^2+a^2}}}



{{{(b^2+a^2)^2}}} Condense the terms.



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Answer:



So {{{b^4+2a^2b^2+a^4}}} factors to {{{(b^2+a^2)^2}}}.



In other words, {{{b^4+2a^2b^2+a^4=(b^2+a^2)^2}}}.



Note: you can check the answer by expanding {{{(b^2+a^2)^2}}} to get {{{b^4+2a^2b^2+a^4}}} or by graphing the original expression and the answer (the two graphs should be identical).


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