Roots of Polynomials

If zero is obtained when a number is substituted for the variable in a polynomial, then that number is a root of the polynomial. For example, 1 is a root of x2 - 1.

Note    A number r is a root of a polynomial if and only if x - r is a factor of that polynomial.

The factorization  

x3 - $\displaystyle {\frac{{8}}{{3}}}$x2 - $\displaystyle {\frac{{5}}{{3}}}$x + 2 = $\displaystyle {\frac{{1}}{{3}}}$$\displaystyle \left(\vphantom{ x-3}\right.$x - 3$\displaystyle \left.\vphantom{ x-3}\right)$$\displaystyle \left(\vphantom{
3x-2}\right.$3x - 2$\displaystyle \left.\vphantom{
3x-2}\right)$$\displaystyle \left(\vphantom{ x+1}\right.$x + 1$\displaystyle \left.\vphantom{ x+1}\right)$

displays the three roots 3, ${\frac{{2}}{{3}}}$, -1. Notice that these roots are precisely the values of the x-coordinate where the graph of y = x3 - ${\frac{{8}}{{3}}}$x2 - ${\frac{{5}}{{3}}}$x + 2 crosses the x-axis.

dtbpFU2.9992in1.9995in0pt y = x3 - ${\frac{{8}}{{3}}}$x2 - ${\frac{{%
5}}{{3}}}$x + 2Plot

The preceding graph was created within Scientific Notebook. Click hereDM6.tex for the chapter discussing techniques for such displays.

The factorization

x3 - $\displaystyle {\frac{{13}}{5}}$ix2 -8x2 + $\displaystyle {\frac{{29}}{5}}$ix + $\displaystyle {\frac{{81}}{5}}$x + 6i - $\displaystyle {\frac{{18}}{5}}$

= $\displaystyle {\textstyle\frac{1}{5}}$$\displaystyle \left(\vphantom{ 5x+2i}\right.$5x + 2i$\displaystyle \left.\vphantom{ 5x+2i}\right)$$\displaystyle \left(\vphantom{ x-5-3i}\right.$x - 5 - 3i$\displaystyle \left.\vphantom{ x-5-3i}\right)$$\displaystyle \left(\vphantom{ x-3}\right.$x - 3$\displaystyle \left.\vphantom{ x-3}\right)$

shows the three roots - ${\frac{2}{5}}$i, 5 + 3i, 3.

You can find all real and complex roots directly by applying Roots from the Polynomials menu.

$\blacktriangleright$ To find the roots of a polynomial

1.
Type the polynomial.

2.
Leave the insertion point in the expression.

3.
From the Polynomials submenu, choose Roots.

$\blacktriangleright$ Polynomials + Roots

x3 - ${\frac{{13}}{{5}}}$ix2 -8x2 + ${\frac{{29}}{{5}}}$ix + ${\frac{{81}}{{5}}}$x + 6i - ${\frac{{18}}{{5}}}$, roots:
- ${\frac{{2}}{{5}}}$i
5 + 3i
3

It follows from the Fundamental Theorem of Algebra that the number of roots (including complex roots and multiplicities) is the same as the degree of the polynomial.

For polynomials with rational (real or complex) coefficients, Scientific Notebook uses the usual formulas for finding roots symbolically for polynomials of degree 4 or less, and it finds the roots numerically for polynomials of higher degree. This behavior is due to the mathematical phenomenon that there is no general formula in terms of radical expressions for the roots of polynomials of degree 5 and higher.



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