Polynomials + Roots
x3 +3x2 + 3x + 1, roots:
-1 -1 -1
x4 +4x3 -12x2 - 32x + 64, roots:
-4 -4 2 2
x3 + cx - 1, roots:,
- +
i
pt
-
i
pt
α =![]()
Factor the polynomials to observe the repeated roots.
Factor
x3 +3x2 +3x + 1 =x + 1
6pt
x4 +4x3 -12x2 -32x + 64 =x + 4
x - 2
6pt
x3 -x - 1 =
x -
2x +
6pt
The roots of third- and fourth-degree polynomials can be complicated, with multiple embedded radicals in the expressions. To put those roots in simpler form, you may want numerical approximations. You get numerical results if you enter at least one coefficient in decimal notation. You can also get a numerical form directly from the symbolic one by applying Evaluate Numerically to the matrix of roots. The following example shows both a symbolic solution and a numerical solution (with Digits Used in Display set to 6).
Polynomials + Roots
x3 + 3x + 1, roots:
- +
6pt
-
+
i
-
-
6pt
-
-
i
-
-
x3 + 3x + 1.0, roots:
- .322185 .161093 - 1.75438i .161093 + 1.75438i
Substituting the exact roots for x in the polynomial x3 + 3x + 1 gives zero, as it should. Applying Evaluate has little effect, but both Simplify and Factor give the following result.
Simplify or Factor
-
+
+3
-
+
+ 1 = 0
Using the numerical approximations to the roots, you get a very small, but nonzero, value. You can get closer approximations to the roots by increasing the number of displayed digits before finding the roots.
Evaluate
- .322185
+3
- .322185
+1.0 = 1.174312318×10-6 6pt
- .32218535462608559291
+3
- .32218535462608559291
+ 1
= 4.870126439×10-21
Polynomials + Roots
x4 - 2x - 3, roots:
-1 6pt -
+
6pt
- +
+
+ i
+
6pt
- +
+
- i
+
You can evaluate the roots numerically or enter a coefficient in floating-point notation before finding roots to get the following.
Polynomials + Roots
x4 - 2x - 3.0, roots:![]()