Andy C. answered • 07/20/18
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x^2 - x - 3 = a(x^2 - 2x + 1) + b(x-1) + c <---- FOIL
= ax^2 -2ax + a + bx - b + c <--- distributive
= ax^2 + (b-2a)x + (a-b+c) <--- CLT
Equating coefficients:
a=1
b -2a = -1
a-b+c = -3
b-2(1) = -1
b - 2 = -1
b = 1
1 - 1 + c = -3
c = -3
So the polynomial of the right is : (x-1)^2 + x-1 + -3 =
x^2 - 2x + 1 + x - 1 - 3 =
x^2 - x -3
a=b=1, c=-3
Andy C.
CLT stands for COMBINE LIKE TERMS....
the coefficients of x and B and -2A
a , -b, and C are the remaining constants
so all that step does is combine them into a single term
just as you would combining like terms
Next the two polynomials on the left and right are the same, so their
coefficients must be the same.
Therefore, A, the coefficient of x^2 on the right MUST be the same as 1,
the coefficient of x^2 on the left.
LIkewise, b-2a, the coefficient of x on teh right MUST be the same as -1,
the coefficient of x on the left
Finally, the constant term on the right a-b+c MUST be the same as the constant term on the left
which is -3.
Equating these coefficients results in the three equations listed.
Solving this 3x3 system of equations results in a=b=1 and c=-3
Plugging them into the polynomial on the right and simplifying
gives the polynomial on the left which proves the answer is correct
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07/20/18
Iskandar K.
07/20/18