How to Equate Coefficients

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Polynomial mathematical expressions comprise a variable denoted "X" in various powers multiplied by numbers called coefficients. For instance, in the polynomial "7X^2+13X-8" the coefficient at "X" in power 2 is 7. Remember that "X" in power of "0" is 1. Equating coefficients is a method of equation solving that equalizes coefficients at "X" in the same power in two polynomials. This method is useful to rearrange polynomial formulas For example, the task is to bring a polynomial "6X-2" to the following form "A(X-1)+B(X+1)." In steps below we illustrate how to equate coefficients and find unknown parameters "A" and "B."

  • Consider the distributive law in mathematics that states: a(b+c)=ab+ac.
    Apply the distributive law to the expression "A(X-1)+B(X+1)."
    A(X-1)+B(X+1)=AX-A+BX+B.

  • Combine the terms having the variable "X" and apply the distributive law again.
    AX-A+BX+B=AX+BX+B-A=X(A+B)+(B-A).
    Remember that the initial polynomial "6X-2" must be equal to "A(X-1)+B(X+1)." Hence it also equals "X(A+B)+(B-A)."

  • Equate coefficients for expression terms containing "X" in the same power. For the expressions "6X-2" and "X(A+B)+(B-A)" coefficients at "X" are "6" and "(A+B)" while at 1 (X^0) they are "-2" and "(B-A)." It leads to a system of the two simultaneous equations.

    6=(A+B) and -2=(B-A).

  • Add up the simultaneous equations.
    6-2=A+B+B-A or 4=2B.
    Divide both sides of the resulting equation by 2, obtain that B=2.

  • Substitute "B" with its value to calculate "A" from any of the equations.
    6=A+2 or A=6-2. Thus ,A=4.

  • In the form "A(X-1)+B(X+1)", substitute "A" and "B" with their values to get
    6X-2= 4(X-1)+2(X+1)."

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