Perfect Square Polynomial: Finding (a+b) for P(x)=x^4+ax^3+bx^2-8x+1

[ritwik06]

Homework Statement

If the polynomial P(x)=x^4+ax^3+bx^2-8x+1 is a perfect square then find (a+b)

The Attempt at a Solution

The first term is a perfect square and so is the last term. that means that th middle terms should have been =2x^2.
But it is not so, then how can this be a perfect square?

 

[HallsofIvy]

P(x) is a perfect square of what? Since it involves x⁴, it must be the square of a quadratic. What is (x^2+ ux+ 1)²? What values of a, b, and u make those the same?

 

[ritwik06]

P(x) is a perfect square of what? Since it involves x^{4[/itex], it must be the square of a quadratic. What is (x^2+ ux+ 1)2? What values of a, b, and u make those the same?}

^{Thanks. I have solved the problem. But will you please tell me how did you figure out that the skeleton expression given was the square of a trinomial?}

 

[HallsofIvy]

I didn't "figure" that out. The given expression involved x⁴ so I knew it must be the square of a quadratic. The most general quadratic is ax²+ bx+ c. Then I saw that, in order to get "x⁴" and "+ 1" I must have a= 1, c= 1. I still did not know what b was so I left that in. I did not know that b was not 0, just that there was no reason to assume it wasn't!