Question

The graph of polynomial
$y=a{x}^4+b{x}^3+c{x}^2+dx+e$ is shown. Find the coefficients $a,b,c,d$ and $e$

Answer 1

This polynomial has a zero of multiplicity $2$ at $x=-2$ and a zero of multiplicity $2$ at $x=2$. Hence it may be written as

$y=a(x+2{)}^2(x-2{)}^2$

We now use the $y$ intercept at $(0,-2)$ to write the equation

$-2=a(0+2{)}^2(0-2{)}^2$

Solve the above for a to obtain

$a=-\frac{1}{8}$

We now write the polynomial as follows

$y=(-\frac{1}{8})(x+2{)}^2(x-2{)}^2$

Expand

$y=(-\frac{1}{8})(x^4-8x^2+16)$

We now identify the coefficients by comparing the polynomial

$y=(-\frac{1}{8})x^4+x^2-2$ by the polynomial $y=a{x}^4+b{x}^3+c{x}^2+dx+e$ we get,

$a=-\frac{1}{8}$, $b=0$, $c=1$, $d=0$, $e=-2$