Method of undetermined coefficients

How to divide a polynomial by a polynomial using the method of undetermined coefficients

To divide a polynomial by a polynomial, in addition to the method of dividing polynomials by "angle", you can use the method of undetermined coefficients. The essence of this method is as follows.
Let it be required to divide a polynomial
a(x)= a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ... +  a₁x + a₀
by a polynomial
b(x)= b_mx^m + b_m-1x^m-1 + b_m-2x^m-2 + ... + b₁x + b₀,
that is, it is required to represent the polynomial a(x) in the form a(x)=b(x)* c(x) + r(x),
where the polynomial a(x) is the dividend, the polynomial b(x) is the divisor, the polynomial c(x) is the quotient, and the polynomial r(x) is the remainder. The degree of the quotient c(x) is equal to the difference between the degrees of the dividend and the divisor, and the degree of the remainder r(x) is less than the degree of the divisor, therefore, the maximum degree of r(x) can be equal to m-1.
Thus, the quotient c(x) is a polynomial of degree n-m with unknown coefficients _i
c(x)= c_n-mx^n-m + c_n-m-1x^n-m-1 + c_n-m-2x^n-m-2 + ... + c₁x + c₀,
and the remainder r(x) is a polynomial of degree m-1 with unknown coefficients r_j
r(x)= r_m-1x^m-1 + r_m-2x^m-2 + r_m-3x^m-3 + ... + r₁x + r₀.
To find the unknown coefficients c_i and r_j, simply multiply b(x)* c(x), add to r(x) and equate the coefficients of the polynomials with the same powers of x on the left and right sides of the equality a(x)=b(x)*c(x) + r(x).
Let us consider examples illustrating the use of the method of undetermined coefficients when dividing a polynomial by a polynomial.

Examples of dividing a polynomial by a polynomial using the method of undetermined coefficients

Example 1.
Divide the polynomial 5x⁴ - 3x³ + 2x² - x + 3 by a polynomial x³ - 2x² + 1 using the method of undetermined coefficients.
Solution.
The dividend a(x)=5x⁴ - 3x³ + 2x² - x + 3 - a polynomial of degree 4,
the divisor b(x)= x³ - 2x² + 1 - a polynomial of degree 3.
Therefore, the quotient c(x) is a polynomial of degree 4-3 = 1
c(x) = c₁x + c₀,
and the remainder r(x) is a polynomial of degree 3-1=2
r(x) = r₂x² + r₁x + r₀.
Multiplying and adding polynomials in an expression b(x)*c(x) + r(x), we get
(x³-2x² + 1)*(c₁x + c₀) + r₂x² + r₁x + r₀ = c₁*x⁴ + x³*(c₀ - 2c₁) + x²*(r₂ - 2c₀) + x*(c₁ + r₁ ) + c₀ + r₀.
By equating the coefficients of the same powers of x in the equality
a(x)=b(x)* c(x) + r(x),
we obtain a system of equations for finding unknowns c₀, c₁, r₀, r₁, r₂.
c₁=5,
c₀ - 2c₁=-3,
r₂ - 2c₀=2,
c₁ + r₁=-1,
c₀ + r₀=3
By sequentially solving the equations using the substitution of known values, _i, r_j, we find the solution to the system
c₁=5,
c₀=7,
r₂=16,
r₁=-6,
r₀=-4.
Hence, c(x) = 5x + 7; r(x)=16x² - 6x - 4.
Answer: 5x⁴ - 3x³ + 2x² - x + 3 = (x³ - 2x² + 1)*(5x + 7) +  16x² - 6x - 4.

Example 2.
Divide the polynomial 6x⁵ - 11x⁴ + 10x³ - 10x² + 3x - 2 by a polynomial 2x³ - 3x² + x - 1 using the method of undetermined coefficients.
Solution.
The dividend a(x) = 6x⁵ - 11x⁴ + 10x³ - 10x² + 3x - 2 - a polynomial of degree 5,
the divisor b(x)= 2x³ - 3x² + x - 1 - a polynomial of degree 3.
Therefore, the quotient c(x) is a polynomial of degree 5-3 = 2
c(x) = c₂x² + c₁x + c₀,
and the remainder r(x) is a polynomial of degree 3-1=2
r(x) = r₂x² + r₁x + r₀.
Multiplying and adding polynomials in an expression b(x)*c(x) + r(x), we get
(2x³ - 3x² + x - 1)*(c₂x² + c₁x + c₀) + r₂x² + r₁x + r₀ =
= 2c₂x⁵ - x⁴(2c₁ - 3c₂) + x³(c₂ - 3c₁ + 2c₀) + x²(c₁ - c₂ - 3c₀ + r₂) + x(c₀ - c₁ + r₁) + r₀ - c₀.
By equating the coefficients of the same powers of x in the equality a(x)=b(x)* c(x) + r(x),
we obtain a system of equations for finding unknowns c₀, c₁, c₂, r₀, r₁, r₂
2c₁ - 3c₂ = -11,
2c₀ - 3c₁ + c₂=10,
r₂ - 3c₀ + c₁ - c₂=-10,
c₀ - c₁ + r₁=3,
r₀ - c₀=-2
By sequentially solving the equations using the substitution of known values, _i, r_j, we find the solution to the system
c₂=3,
c₁=-1,
c₀=2,
r₂=0,
r₁=0,
r₀=0.
Hence, c(x)=3x² - x + 2; r(x)=0.
Answer: 6x⁵ - 11x⁴ + 10x³ - 10x² + 3x - 2 = (2x³ - 3x² + x - 1)*(3x² - x + 2).