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

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In summary, the polynomial P(x)=x^4+ax^3+bx^2-8x+1 is a perfect square if it is the square of a quadratic. To find (a+b), set up the expression (x^2+ ux+ 1)^2 and solve for the values of a, b, and u that make it equivalent to P(x).
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ritwik06
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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?
 
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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?
 
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  • #3
HallsofIvy said:
P(x) is a perfect square of what? Since it involves x4[/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?
 
  • #4
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!
 

1. What is a perfect square?

A perfect square is a number that is the result of multiplying a whole number by itself. For example, 9 is a perfect square because it is the product of 3 multiplied by 3.

2. How do you know if a number is a perfect square?

A number is a perfect square if its square root is a whole number. For example, the square root of 25 is 5, making 25 a perfect square.

3. What are some examples of perfect squares?

Some examples of perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc. Any whole number multiplied by itself will result in a perfect square.

4. Can negative numbers be perfect squares?

No, negative numbers cannot be perfect squares. A perfect square must have a positive square root, and the square root of a negative number is imaginary.

5. How are perfect squares useful in mathematics?

Perfect squares are useful in a variety of mathematical concepts, such as finding the area of squares and rectangles, solving equations, and identifying patterns in numbers. They are also commonly used in geometry and algebra.

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