Chapter 4.2, Problem 62E

To determine

To show: that the coefficient of $x^3$ is the sum of $a,b,c$ and $d$ , and the constant term is the product of $a,b,c$ and $d$ .

Explanation of Solution

Given information:

The function

  $f\left(x\right)=\left(x+a\right)\left(x+b\right)\left(x+c\right)\left(x+d\right)$

When, $f\left(x\right)$ s written in standard form

Calculation:

Consider the function $f\left(x\right)=\left(x+a\right)\left(x+b\right)\left(x+c\right)\left(x+d\right)$

Multiply first two terms and last two terms together.

  $f\left(x\right)=\left(x^2+ax+bx+ab\right)\left(x^2+cx+dx+cd\right)$

Now multiply each term of first parenthesis to the each term of second parenthesis.

  $\begin{array}{c}f\left(x\right)=x^2\left(x^2+cx+dx+cd\right)+ax\left(x^2+cx+dx+cd\right)+bx\left(x^2+cx+dx+cd\right)+ab\left(x^2+cx+dx+cd\right)\\ =x^4+c{x}^3+d{x}^3+cd{x}^2+a{x}^3+ac{x}^2+ad{x}^2+acdx+b{x}^3+bc{x}^2+bd{x}^2+bcdx+ab{x}^2+abcx+abdx+abcd\\ =x^4+\left(a{x}^3+b{x}^3+c{x}^3+d{x}^3\right)+\left(cd{x}^2+ac{x}^2+ad{x}^2+bc{x}^2+bd{x}^2+ab{x}^2\right)+\left(acdx+bcdx+abcx+abdx\right)+abcd\\ =x^4+\left(a+b+c+d\right)x^3+\left(cd+ac+ad+bc+bd+ab\right)x^2+\left(acd+bcd+abc+abd\right)x+abcd\end{array}$

Thus, the coefficient of $x^3$ is sum of $a,b,c,d$ and constant term is product of $a,b,c,d$ .