Home » Blog » Use Bolzano’s theorem to isolate real roots of given polynomials

Use Bolzano’s theorem to isolate real roots of given polynomials

by RoRi

A real value x is a root of a function f if f(x) = 0. We say we have isolated a real root if we find an interval [a,b] such that x \in [a,b] and no other real roots are in the interval. Use Bolzano’s Theorem to isolate the real roots of the following:

  1. 3x^4 - 2x^3 - 36x^2+ 36x - 8 = 0.
  2. 2x^4 - 14x^2 + 14x -1 = 0.
  3. x^4 + 4x^3 + x^2 - 6x + 2 = 0.
  1. We have

        \begin{alignat*}{4} f(-4) &= 168, \quad &f(-3) &=& -143 &\qquad \implies \qquad & f(c_1) &=& 0 \quad \text{for} \quad c_1 \in [-4,-3].\\ f(0) &= -8, \quad &f(0.5) &=& \frac{15}{16} &\qquad \implies \qquad & f(c_2) &=& 0 \quad \text{for} \quad c_2 \in [0,0.5]\\ f(0.5) &= \frac{15}{16}, \quad &f(1) &=& -7 &\qquad \implies \qquad & f(c_3) &=& 0 \quad \text{for} \quad c_3 \in [0.5,1] \\ f(1) &= -7, \quad &f(4) &=& 200 &\qquad \implies \qquad & f(c_4) &=& 0 \quad \text{for} \quad c_4 \in [1,4] \end{alignat*}

  2. We have

        \begin{alignat*}{4} f(-4) &= 231, \quad &f(-3) &=& -7 &\qquad \implies \qquad & f(c_1) &=& 0 \quad \text{for} \quad c_1 \in [-4,-3].\\ f(0) &= -1, \quad &f(1) &=& 1 &\qquad \implies \qquad & f(c_2) &=& 0 \quad \text{for} \quad c_2 \in [0,1]\\ f(1) &= 1, \quad &f(1.5) &=& -\frac{11}{8} &\qquad \implies \qquad & f(c_3) &=& 0 \quad \text{for} \quad c_3 \in [1,1.5] \\ f(1.5) &= -\frac{11}{8}, \quad &f(2) &=& 3 &\qquad \implies \qquad & f(c_4) &=& 0 \quad \text{for} \quad c_4 \in [1.5,2] \end{alignat*}

  3. We have

        \begin{alignat*}{4} f(-3) &= 2, \quad &f(-2.5) &=& -\frac{3}{16} &\qquad \implies \qquad & f(c_1) &=& 0 \quad \text{for} \quad c_1 \in [-3,-2.5].\\ f(-2.5) &= -\frac{3}{16}, \quad &f(-2) &=& 2 &\qquad \implies \qquad & f(c_2) &=& 0 \quad \text{for} \quad c_2 \in [-2.5,-2]\\ f(0) &= 2, \quad &f(0.5) &=& -\frac{3}{16} &\qquad \implies \qquad & f(c_3) &=& 0 \quad \text{for} \quad c_3 \in [0,0.5] \\ f(0.5) &= -\frac{3}{16}, \quad &f(1) &=& 2 &\qquad \implies \qquad & f(c_4) &=& 0 \quad \text{for} \quad c_4 \in [0.5,1] \end{alignat*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):