the sum of two polynomials is 40x^5-2x^4+x^3-7x^2-1. if the polynomial is the square of (x^2-7x+1), find the other polynomial
"the sum of two polynomials is 40x^5-2x^4+x^3-7x^2-1. if one polynomial is the square of (x^2-7x+1), find the other polynomial"
Square by find the sum of
1. squares of individual terms,
2. 2 times the product of cross terms
P1=(x^2-7x+1)²
=x^4+49x²+1 -14x^3 -14x + 2x^sup2;
= x^4-14x^3+51x^2-14x+1
Since the given polynomial
P=40x^5-2x^4+x^3-7x^2-1
is the sum of two polynomials P1 and P2, we can find P2 by subtracting P1 from P.
Remember the subtraction can only be done on the coefficients of like terms.
P-P1
=40x^5-2x^4+x^3-7x^2-1 - (x^4-14x^3+51x^2-14x+1)
=40x^5 +(-2x^4-x^4) + (x^3+14x^3) + (-7x^2-51x^2) + 14x + (-1-1)
= ....
Can you take it from here?
To find the other polynomial, we need to start by factoring the given polynomial, which is the sum of two polynomials. Let's call the other polynomial we're trying to find Y. The given polynomial is:
40x^5 - 2x^4 + x^3 - 7x^2 - 1
We are also given that this polynomial is the square of (x^2 - 7x + 1). So, we can set up the equation:
(x^2 - 7x + 1)^2 = 40x^5 - 2x^4 + x^3 - 7x^2 - 1
To solve this equation, we will expand the square on the left side:
(x^2 - 7x + 1)^2 = (x^2 - 7x + 1)(x^2 - 7x + 1)
= x^4 - 14x^3 + x^2 - 14x^3 + 49x^2 - 7x - x^2 + 7x - 1
= x^4 - 14x^3 + 50x^2
Now that we have the expanded form of (x^2 - 7x + 1)^2, we can compare it to the given polynomial:
x^4 - 14x^3 + 50x^2 = 40x^5 - 2x^4 + x^3 - 7x^2 - 1
By rearranging terms and combining like terms, we get:
40x^5 - 3x^4 - 13x^3 + 43x^2 + 1 = 0
Now, this equation represents the sum of the two polynomials. To find the other polynomial, we need to subtract the square of (x^2 - 7x + 1) from this equation:
Y = 40x^5 - 3x^4 - 13x^3 + 43x^2 + 1 - (x^4 - 14x^3 + 50x^2)
Simplifying the expression, we get:
Y = 40x^5 - 3x^4 - 13x^3 + 43x^2 + 1 - x^4 + 14x^3 - 50x^2
Combining like terms, we finally find the other polynomial:
Y = 40x^5 - 4x^4 + x^3 - 7x^2 + 1