Question 2.32: Given that a + b is a factor of a³ + b³, find all remaining ...
Given that a + b is a factor of a³ + b³, find all remaining factors.
We can approach this problem using long division, since the factors of any expression when multiplied together produce that expression. So we can determine the factors using the inverse of multiplication: that is, division. Now, we are dividing by two literal numbers a and b, so starting with the unknown a,we see that a divides into a³. Think of it as 3 into 27, leaving 9 or 3², then a into a³ is a². Another approach is simply to apply the laws of indices: a³/a¹ = a², thus a¹ and a² are factors of a³. This first part of the division is shown below:
\left.a+b\begin{matrix} \\ \ \end{matrix} \right)\overset{ \ \ \ \ \ \ \ \ a^2}{ \overline{\begin{matrix} a^3+b^3 \\ a^3+a^2b\\\hline\end{matrix} } } \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad -a^2b+b^3\text{(after subtruction)}
Notice that the second row underneath the division is obtained by multiplying the divisor (the expression doing the dividing, a + b in our case) by the quotient (the result above the division line, a² in our case). The remainder is obtained after subtraction of the second row from the original expression.
Next we need to find a quotient which when multiplied by the divisor gives us −a²b (the first term in the bottom line). We hope you can see that −ab when multiplied by the first term in the divisor a gives us −a²b, then −ab is the next term in our quotient, as shown below:
\begin{matrix} \\ \left.a + b\right) \\ \\ \\ \\ \\ \end{matrix}\overset{ a^2 – ab}{ \overline{\begin{matrix} a^3+b^3 \\ a^3+a^2b\\\hline -a^2 b + b^3 \\ -a^2 b – ab^2 \\ \hline\end{matrix}}}\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad +ab^2 + b^3\text{(again aftersubtraction)}
Finally we need the next term in our quotient to yield + ab², when multiplied by the first term of our divisor a. Again, we hope you can see that this is b². This completes the division, as shown below:
\begin{matrix} \\ \left.a + b\right) \\ \\ \\ \\ \\ \\ \\ \end{matrix}\overset{ a^2 – ab + b^2}{ \overline{\begin{matrix} a^3+b^3 \\ a^3+a^2b\\\hline -a^2 b + b^3 \\ -a^2 b – ab^2 \\ \hline \ \ \ \ + ab^2 + b^3 \\ \ \ \ \ + ab^2 + b^3 \\ \hline \end{matrix} } } \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0 \ \text{(after subtrac-tion the remainder is zero)}
The factors of the expression a³ + b³ are therefore: (a + b) and (a² − ab + b²).