Chapter 2.5, Problem 102E

To determine

To find : all the zeros of the function

Answer to Problem 102E

The zeros of the function are: $\boxed{\boxed{x=\frac{2}{3};\sqrt{5}i;-\sqrt{5}i}}$

Explanation of Solution

Given information: $f\left(x\right)=3x^3-2x^2+15x-10$

Concept Involved:

Synthetic Division (for a Cubic Polynomial):To divide $a{x}^3+b{x}^2+cx+d$ by $x-k$ , use this pattern.

$\begin{array}{l}\begin{array}{l}k\\ \end{array}|\stackrel{Coefficientofdividend}{\overbrace{\underset{\_}{\begin{array}{cccc}\boxed{a}& b& c& d\\ \downarrow & \underset{\downarrow }{\boxed{ka}}& \underset{\downarrow }{\boxed{}}& \underset{\downarrow }{\boxed{}}\end{array}}}}\\ \begin{array}{ccccc}& \nearrow & \nearrow & \nearrow & \end{array}\\ \underset{Coefficientofquotient}{\underbrace{\begin{array}{ccc}\boxed{a}& \boxed{}& \boxed{}\end{array}}}\begin{array}{cc}\boxed{r}& \leftarrow \text{Remainder}\end{array}\end{array}$

In case when we have a polynomial with a missing term, insert placeholders with zero coefficients for missing powers of the variable. Vertical pattern: Add terms in columns Diagonal pattern: Multiply results by k. This algorithm for synthetic division works only for divisors of the form x - k. Remember that $x+k=x-\left(-k\right)$ .

The Division Algorithm: If $f\left(x\right)$  and   $d\left(x\right)$ are polynomials such that $d\left(x\right)\ne 0$ , and the degree of $d\left(x\right)$ isless than or equal to the degree of $f\left(x\right)$ , then there exist unique polynomials $q\left(x\right)$  and   $r\left(x\right)$ such that

  $\underset{\stackrel{\uparrow }{{\displaystyle Dividend}}}{{\displaystyle f\left(x\right)}}\text{ = }\underset{\stackrel{\uparrow }{{\displaystyle Divisor}}}{{\displaystyle d\left(x\right)}}\cdot \underset{\stackrel{\uparrow }{{\displaystyle Quotient}}}{{\displaystyle q\left(x\right)}}+\underset{\stackrel{\uparrow }{{\displaystyle Remainder}}}{{\displaystyle r\left(x\right)}}$

…where $r\left(x\right)=0$ or the degree of $r\left(x\right)$ is less than the degree of $d\left(x\right)$ .

If theremainder $r\left(x\right)$ is zero, then $d\left(x\right)$ divides evenly into $f\left(x\right)$ and k can be called as zero of the polynomial.

Calculation:

To find the zeros of the polynomial set the function equal to zero $f\left(x\right)=0$

  $3x^3-2x^2+15x-10=0$

Factor the GCF of first two term and GCF of last two terms

  $x^2\left(3x-2\right)+5\left(3x-2\right)=0$

Factor the common binomial in the left side of the equation

  $\left(3x-2\right)\left(x^2+5\right)=0$

We need to set each factor equal to zero using the zero factor property which states that:

“If $ab=0$ , then $a=0$ or $b=0$ .”

  $x^2+5=0$  ▸1^st equation

  $3x-2=0$  ▸2^nd equation

Solve the 1st equation $x^2+5=0$

• By subtracting 5 on both sides, then
• By Splitting the number inside the square root in the right side of the equation, then
• By taking square root on both sides, then
• By simplifying square root on both sides of the equation, and then
• By replacing $i$ for $\sqrt{-1}$

  $x^2+5-5=0-5$

  $x^2=-5$

  $\sqrt{x^2}=\sqrt{-5}$

  $\sqrt{x^2}=\sqrt{-1}\cdot \sqrt{5}$

  $x=\sqrt{-1}\cdot \pm \sqrt{5}$

  $x=\pm \sqrt{5}i$

Solving the 2nd equation $3x-2=0$

• By adding 2 on both sides, then
• By simplifying the equation on both sides, then
• By dividing 3 on both sides, and then
• By simplifying fraction on both sides of the equation

  $3x-2+2=0+2$

  $3x=2$

  $\frac{3x}{3}=\frac{2}{3}$

  $x=\frac{2}{3}$

Conclusion:

The zeros of the given function $f\left(x\right)=3x^3-2x^2+15x-10$ are: $x=\frac{2}{3};\sqrt{5}i;-\sqrt{5}i$