Chapter 6.8, Problem 18PT

To determine

To calculate: The remaining factors of polynomial $2x^3+15x^2+22x-15$ when it divided by $x+5$ .

Answer to Problem 18PT

The remaining factor of polynomial $2x^3+15x^2+22x-15$ other than $x+5$ are $\left(x+3\right)\left(x-\frac{1}{2}\right)$ .

Explanation of Solution

Given information:

The polynomial is given as:

  $2x^3+15x^2+22x-15$ and the dividend $x+5$ .

Formula used:

The synthetic division method,

Step 1: write the coefficients of the divided so that the degrees of the terms are in descending order. Then write the constant $r$ of the divisor $x-r$ in the box. Write the first coefficient down.

Step 2: Multiply the first coefficient by $r$ and write the product the second coefficient.

Step 3: Add the product and the second coefficient.

Step 4: Again, repeat Step 2 and 3 until reach a sum in last column. The number along the bottom row are the coefficient of the quotient. The final number is remainder.

The quadratic formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The Factor theorem: the binomial $x-r$ is a factor of the polynomial $p\left(x\right)$ if and only if

  $p\left(r\right)=0$

Calculation:

Consider the polynomial,

  $2x^3+15x^2+22x-15$

The binomial $x+5$ is a factor of the polynomial if $-5$ is a zero of the polynomials.

Using the factor theorem and synthetic division method as,

Step 1: Write the coefficient of dividend. The constant $r=-5$ and bring the first coefficient 2 down.

  $$

  $\begin{array}{l}\overline{)-5}2152215\\ \downarrow \\ \overline{2}\end{array}$

Step 2: Multiply the first coefficient by $r$ : $-5\cdot 2=-10$ .the product write under the second coefficient.

  $\begin{array}{l}\overline{)-5}2152215\\ -10\\ \overline{2\nearrow }\end{array}$

Step 3: now write the sum of product and the second coefficient: $15+\left(-10\right)=5$ .

  $\begin{array}{l}\overline{)-5}2152215\\ -10\\ \overline{25}\end{array}$

Step 4: Multiply the sum,5 by $r:-5\cdot 5=-25$ .The product value write under the next coefficient and add: $-25+22=-3$ .Now multiply the sum , $-3$ by $r:-5\cdot -3=15$ . The product value writes under the next coefficient and subtract: $15-15=0$ .

  $\begin{array}{l}\overline{)-5}2152215\\ -10-2515\\ \overline{25\nearrow -3\nearrow 0}\end{array}$

The remainder is 0 so, $x+5$ is a factor of the polynomial.

The polynomial can be factored as $\left(x+5\right)\left(2x^2+5x-3\right)$ .The quotient $\left(2x^2+5x-3\right)$ is in quadratic from so apply the quadratic formula as,

  $\begin{array}{l}x=\frac{-5\pm \sqrt{5^2-4\cdot 2\cdot -3}}{2\cdot 2}\\ x=\frac{-5\pm 7}{4}\\ x=-3,\frac{1}{2}\end{array}$

So, the factors of $\left(2x^2+5x-3\right)$ are $\left(x+3\right)\left(x-\frac{1}{2}\right)$

Thus, the factors of $2x^3+15x^2+22x-15$ are $\left(x+3\right)\left(x-\frac{1}{2}\right)\left(x+5\right)$