Find the values of m and n such that x^4+6x^3+13x^2+mx+n , is a perfect square
we must have squared an expression such as
(x^2 + bx + c)
let's see what we get:
(x^2 + bx + c)^2
= (x^2 + bx + c)(x^2 + bx + c)
= x^4 + bx^3 + cx^2 + bx^3 + b^2x^2 + bcx +cx^2 + 2bcx + c^2
= x^4 + 2bx^3 + x^2(c+b^2+c) + 2bcx + c^2
comparing this with
x^4+6x^3+13x^2+mx+n , we have
2b=6 , ...... 2c +b^2 = 13 ..... 2bc = m ... and c^2 = n
b = 3 ............ c = 2 ........... m=12 ........ n=4
(I started with b=3 and the then substituted across the line
So if m=12 and n=4
we should end up with
(x^2 + 3x + 2)^2
= x^4+6x^3+13x^2+12x+n4
as seen by
http://www.wolframalpha.com/input/?i=expand+%28x%5E2+%2B+3x+%2B+2%29%5E2
To find the values of m and n such that the given expression is a perfect square, we need to determine the conditions for a polynomial to be a perfect square.
A polynomial is a perfect square if and only if the square root of the polynomial is also a polynomial. So, we need to take the square root of the given polynomial and rewrite it in the form of (ax^2 + bx + c)^2.
Let's begin by taking the square root of the given polynomial:
√(x^4 + 6x^3 + 13x^2 + mx + n)
Now, let's try to rewrite this in the form of (ax^2 + bx + c)^2:
(ax^2 + bx + c)^2 = a^2x^4 + 2abx^3 + (2ac + b^2)x^2 + 2bcx + c^2
Comparing the coefficients of like terms from both expressions, we can equate them to each other:
a^2 = 1 ----(1)
2ab = 6 ----(2)
2ac + b^2 = 13 ----(3)
2bc = m ----(4)
c^2 = n ----(5)
From equation (1), we know that a = ±1 since the coefficient of x^4 is 1. Let's consider these two cases separately:
Case 1: a = 1
Using equation (2) to solve for b:
2(1)b = 6
2b = 6
b = 3
Using equation (3) to solve for c:
2(1)c + 3^2 = 13
2c + 9 = 13
2c = 13 - 9
2c = 4
c = 2
Using equation (4) to solve for m:
2(3)(2) = m
12 = m
Using equation (5) to solve for n:
c^2 = n
2^2 = n
n = 4
Therefore, for the case when a = 1, the values of m and n that make the given polynomial a perfect square are m = 12 and n = 4.
Case 2: a = -1
Using equation (2) to solve for b:
2(-1)b = 6
-2b = 6
b = -3
Using equation (3) to solve for c:
2(-1)c + (-3)^2 = 13
-2c + 9 = 13
-2c = 13 - 9
-2c = 4
c = -2
Using equation (4) to solve for m:
2(-3)(-2) = m
12 = m
Using equation (5) to solve for n:
c^2 = n
(-2)^2 = n
n = 4
Therefore, for the case when a = -1, the values of m and n that make the given polynomial a perfect square are m = 12 and n = 4.
In both cases, the values of m and n that make the polynomial x^4 + 6x^3 + 13x^2 + mx + n a perfect square are m = 12 and n = 4.