Chapter 7.6, Problem 41E

To determine

To find: The values of $t$ for which $2x^2+tx+10$ can be written as the product of two binomials.

Answer to Problem 41E

The required values of $t$ are $-12,12$ .

Explanation of Solution

Given information: A polynomial $2x^2+tx+10$ is given.

Calculation: A polynomial of the type $a{x}^2+bx+c$

  $(a\ne 1)$ can be written as the product of two binomials if it is possible to find $p$ and $q$ such that $p+q=\frac{b}{a}$ and $pq=\frac{c}{a}$ .

So, $2x^2+tx+10$ can be written as the product of two binomials if $p$ and $q$ can be found such that $p+q=\frac{t}{2}$ and $pq=\frac{10}{2}=5$ .

A table showing the possible values of $p$ and $q$ and hence the possible values of $t$ is:

  $\begin{array}{lllll}p\hfill & q\hfill & pq\hfill & \frac{p+q=\frac{t}{2}}{}\hfill & t=2(p+q)\hfill \\ 1\hfill & 5\hfill & 5\hfill & 6\hfill & 12\hfill \\ -1\hfill & -5\hfill & 5\hfill & -6\hfill & -12\hfill \end{array}$

Thus, the possible values of $t$ are $-12,12$ and the corresponding products are $(2x-10)(x-1)$ and $(2x+10)(x+1)$ respectively.