Chapter MPS, Problem 6.10MPS

To determine

To Find: The roots and type of roots of given polynomial.

Answer to Problem 6.10MPS

  $0,2or1$

Real and distinct roots

Explanation of Solution

Given:

  $\begin{array}{l}f\left(x\right)=4x^3+4x^2-8x\\ x=time\end{array}$

Formula Used:

  $\begin{array}{l}D=abcd-4b^{3}d+b^2{c}^2-4a{c}^3-27a^2{d}^2\\ D=0\text{Real and one repeated roots}\\ D<0\text{Pair of complex conjugate and one real root}\\ D>0\text{Three real and distinct root}\text{.}\end{array}$

Calculation:

  $\begin{array}{l}f\left(x\right)=4x^3+4x^2-8x\\ x=time\\ a=4\\ b=4\\ c=-8\\ d=0\\ D=0-0+256+1004-0=1260>0\\ \text{Real and distinct roots}\\ \text{⇒}4x^3+4x^2-8x=0\\ \text{⇒}4x\left(x^2+x-2\right)=0\\ \text{⇒}4x\left(x^2-2x+x-2\right)=0\\ \text{⇒}4x\left[x\left(x-2\right)+1\left(x-2\right)\right]=0\\ \text{⇒}4x\left[\left(x+1\right)\left(x-2\right)\right]=0\\ \text{⇒}x=0,2or1\end{array}$

Conclusion:

Thus the roots are real but distinct $0,2or1$ .