Factoring Polynomials

The ability to factor polynomials is an important algebraic tool. You will find that the factoring capabilities of Scientific Notebook are powerful and useful. You can factor polynomials with integer or rational roots and with other roots directly related to the coefficients of the expanded polynomial.

To factor a polynomial, you must type it without using decimal notation. Numbers such as 1.5 are interpreted as floating-point numbers, and Factor does not handle polynomials with floating-point coefficients. Replace decimal numbers with fractions (such as 1.5 = ${\frac{{15}}{{10}}}$), and then apply Factor.


\begin{example} Apply \textsf{Factor } \index{Factor@Factor!polynomial@polyno... ...\left( x+3i\right) ^{2}\left( 12x-3+8i\right) ^{2}\vspace{14pt}$ \end{example}

Note that factoring polynomials generally returns factors with integer coefficients. Factor is effective primarily for polynomials with integer or rational coefficients, although it also factors polynomials whose roots are closely related to the coefficients, as demonstrated in the two preceding examples. Technically, the polynomial is factored over the field generated by its coefficients. If all the coefficients are rational, then the polynomial is factored over the rationals. Apply Factor to get the following product.

$\blacktriangleright$ Factor

${\frac{{1}}{{2}}}$x2 +3x - ${\frac{{20}}{{9}}}$ =  ${\frac{{1}}{{18}}}$$\left(\vphantom{ 3x+20}\right.$3x + 20$\left.\vphantom{ 3x+20}\right)$$\left(\vphantom{ 3x-2}\right.$3x - 2$\left.\vphantom{ 3x-2}\right)$

You can factor not only the difference of two squares and the sum and difference of two cubes, but also the difference of any two equal powers with Scientific Notebook.

$\blacktriangleright$ Factor

x2 - y2 = $\left(\vphantom{ x-y}\right.$x - y$\left.\vphantom{ x-y}\right)$$\left(\vphantom{ x+y}\right.$x + y$\left.\vphantom{ x+y}\right)$

x3 - y3 = $\left(\vphantom{ x-y}\right.$x - y$\left.\vphantom{ x-y}\right)$$\left(\vphantom{ x^{2}+xy+y^{2}}\right.$x2 + xy + y2$\left.\vphantom{ x^{2}+xy+y^{2}}\right)$

x4 - y4 = $\left(\vphantom{ x-y}\right.$x - y$\left.\vphantom{ x-y}\right)$$\left(\vphantom{ x+y}\right.$x + y$\left.\vphantom{ x+y}\right)$$\left(\vphantom{ x^{2}+y^{2}}\right.$x2 + y2$\left.\vphantom{ x^{2}+y^{2}}\right)$

You can also factor the sum of any two equal odd powers.

$\blacktriangleright$ Factor

x3 + y3 = $\left(\vphantom{ x+y}\right.$x + y$\left.\vphantom{ x+y}\right)$$\left(\vphantom{ x^{2}-xy+y^{2}}\right.$x2 - xy + y2$\left.\vphantom{ x^{2}-xy+y^{2}}\right)$

x5 + y5 = $\left(\vphantom{ x+y}\right.$x + y$\left.\vphantom{ x+y}\right)$$\left(\vphantom{ x^{4}-x^{3}y+x^{2}y^{2}-xy^{3}+y^{4}}\right.$x4 - x3y + x2y2 - xy3 + y4$\left.\vphantom{ x^{4}-x^{3}y+x^{2}y^{2}-xy^{3}+y^{4}}\right)$

x7 + y7 = $\left(\vphantom{ x+y}\right.$x + y$\left.\vphantom{ x+y}\right)$$\left(\vphantom{ y^{6}-xy^{5}+x^{2}y^{4}-x^{3}y^{3}+x^{4}y^{2}-x^{5}y+x^{6}}\right.$y6 - xy5 + x2y4 - x3y3 + x4y2 - x5y + x6$\left.\vphantom{ y^{6}-xy^{5}+x^{2}y^{4}-x^{3}y^{3}+x^{4}y^{2}-x^{5}y+x^{6}}\right)$