1. ## Polynomial function

If $\displaystyle f(x)=x^4+2x^3+5x^2-16x-20$ , show that f(x) can be expressed in the form $\displaystyle (x^2+x+a)^2-4(x+b)^2$ .

I don really know how to show , but i do know how to determine the value of a and b . Is there a way ?

If $\displaystyle f(x)=x^4+2x^3+5x^2-16x-20$ , show that f(x) can be expressed in the form $\displaystyle (x^2+x+a)^2-4(x+b)^2$ .

I don really know how to show , but i do know how to determine the value of a and b . Is there a way ?
Expand the second one and then equate the coefficients of like powers of x.

3. If what your first post says is correct in your saying that you have determined values of such a and b, you have proved the statement.

4. ## Re :

If what your first post says is correct in your saying that you have determined values of such a and b, you have proved the statement.

Did u mean that by just finding the values of a and b is sufficient to prove that statement ?

If $\displaystyle f(x)=x^4+2x^3+5x^2-16x-20$ , show that f(x) can be expressed in the form $\displaystyle (x^2+x+a)^2-4(x+b)^2$ .

I don really know how to show , but i do know how to determine the value of a and b . Is there a way ?
$\displaystyle (x^2+x+a)^2-4(x+b)^2$ .

$\displaystyle =x^4+x^2 + a^2 + 2x^3 + 2ax + 2ax^2 -4x^2-4b^2 -8xb$

$\displaystyle =x^4 + 2x^3+(2a-3)x^2 + (2a-8b)x +(a^2 -4b^2)$

Comparing above eqn. with

$\displaystyle f(x)=x^4+2x^3+5x^2-16x-20$

$\displaystyle 2a-3 = 5$

$\displaystyle \implies a=4$

$\displaystyle 2a-8b= -16$

$\displaystyle \implies b= 3$

You can check by putting the value
$\displaystyle a^2-4b^2 = -20$
This is what you did

Hence you have shown that values of a and b are real , proving that statement is absolutely correct