Tenth Grade Mathematics: Division Algorithm for Polynomials
Just as the division of whole numbers produces a quotient and a smaller remainder, dividing one polynomial by another partitions algebraic expressions according to their powers. The Division Algorithm for Polynomials establishes that any polynomial can be divided by a non-zero divisor polynomial to yield a unique quotient and a remainder whose degree is strictly less than the divisor. This algorithm provides the foundational mechanism for factoring higher-degree polynomials and identifying remaining roots when some zeros are already known.
Statement of the Division Algorithm
The Division Algorithm for polynomials formalizes the relationship between the dividend, divisor, quotient, and remainder.
Theorem: If P(x) and g(x) are any two polynomials with g(x) != 0, then we can find
unique polynomials q(x) and r(x) such that:
P(x) = g(x) * q(x) + r(x)
where either r(x) = 0 or deg(r(x)) < deg(g(x)).
Component Summary:
P(x) = Dividend polynomial (degree n)
g(x) = Divisor polynomial (degree m, where m <= n)
q(x) = Quotient polynomial (degree n - m)
r(x) = Remainder polynomial (degree < m, or r(x) = 0)
If r(x) = 0, the divisor g(x) divides P(x) completely, meaning g(x) is a factor of P(x).
Step-by-Step Long Division of Polynomials
Performing polynomial long division mirrors integer division, focusing on eliminating the highest remaining power at each successive step:
Arrange both dividend P(x) and divisor g(x) in standard form with descending powers of x. If any intermediate power is missing, include it with a coefficient of zero (such as 0*x^2) as a placeholder. Divide the leading term of the dividend by the leading term of the divisor to obtain the first term of the quotient. Multiply the entire divisor by this new quotient term and subtract the resulting polynomial from the dividend. Bring down the remaining terms to form the new dividend. Repeat this cycle until the degree of the new dividend is strictly less than the degree of the divisor. The remaining expression is r(x).
Worked Division Example:
Divide P(x) = 2*x^2 + 3*x + 1 by g(x) = x + 2.
2*x - 1 <--- Quotient q(x)
+---------------------
x + 2 | 2*x^2 + 3*x + 1
- (2*x^2 + 4*x) <--- (2*x) * (x + 2)
---------------------
-x + 1
- (-x - 2) <--- (-1) * (x + 2)
---------------------
3 <--- Remainder r(x)
Verification:
g(x)*q(x) + r(x) = (x + 2)*(2*x - 1) + 3
= 2*x^2 - x + 4*x - 2 + 3
= 2*x^2 + 3*x + 1 = P(x) (Verified)
Finding All Zeros When Given Specific Zeros
A powerful application of the division algorithm arises when finding all zeros of a higher-degree polynomial when two or more zeros are already provided.
General Solution Procedure:
1. If alpha and beta are given zeros, form the quadratic factor:
g(x) = (x - alpha)*(x - beta) = x^2 - (alpha + beta)*x + alpha*beta.
2. Divide the original higher-degree polynomial P(x) by g(x).
Because alpha and beta are zeros, the remainder r(x) must equal 0.
3. Factor the resulting quotient polynomial q(x) to obtain the remaining zeros.
Advanced Worked Example: Quartic Polynomial
Find all the zeros of 2x^4 - 3x^3 - 3x^2 + 6x - 2, given that two of its zeros are sqrt(2) and -sqrt(2).
Step 1: Form the divisor polynomial from the given zeros:
g(x) = (x - sqrt(2)) * (x - (-sqrt(2)))
= (x - sqrt(2)) * (x + sqrt(2))
= x^2 - 2
Step 2: Divide P(x) = 2*x^4 - 3*x^3 - 3*x^2 + 6*x - 2 by (x^2 - 2):
Dividing leading terms: (2*x^4) / (x^2) = 2*x^2.
Subtract: (2*x^4 - 3*x^3 - 3*x^2) - (2*x^4 - 4*x^2) = -3*x^3 + x^2 + 6*x - 2.
Dividing leading terms: (-3*x^3) / (x^2) = -3*x.
Subtract: (-3*x^3 + x^2 + 6*x) - (-3*x^3 + 6*x) = x^2 - 2.
Dividing leading terms: (x^2) / (x^2) = +1.
Subtract: (x^2 - 2) - (x^2 - 2) = 0.
Quotient obtained: q(x) = 2*x^2 - 3*x + 1.
Remainder: r(x) = 0.
Step 3: Find the remaining zeros by setting q(x) = 0:
2*x^2 - 3*x + 1 = 0
2*x^2 - 2*x - x + 1 = 0
2*x*(x - 1) - 1*(x - 1) = 0
(x - 1)*(2*x - 1) = 0
x = 1 or x = 1/2.
Conclusion:
The four zeros of the quartic polynomial are sqrt(2), -sqrt(2), 1, and 1/2.
Chapter Exercises
Exercise 1: Divide P(x) = 3x^3 + x^2 + 2x + 5 by g(x) = 1 + 2*x + x^2. State the quotient and the remainder, and explicitly verify the division algorithm identity.
Exercise 2: Find all the zeros of the polynomial P(x) = x^4 - 6x^3 - 26x^2 + 138*x - 35, given that two of its zeros are 2 + sqrt(3) and 2 - sqrt(3).
Exercise 3: On dividing x^3 - 3x^2 + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and -2x + 4, respectively. Find the polynomial g(x).
Solutions and Explanations
Solution 1: Write both polynomials in standard form: P(x) = 3x^3 + x^2 + 2x + 5 and g(x) = x^2 + 2x + 1. Dividing the leading terms: (3x^3)/(x^2) = 3x. Multiplying g(x) by 3x: 3x^3 + 6x^2 + 3x. Subtracting from P(x): (3x^3 + x^2 + 2x + 5) - (3x^3 + 6x^2 + 3x) = -5x^2 - x + 5. Dividing leading terms: (-5x^2)/(x^2) = -5. Multiplying g(x) by -5: -5x^2 - 10x - 5. Subtracting: (-5x^2 - x + 5) - (-5x^2 - 10x - 5) = 9x + 10. The quotient is q(x) = 3x - 5, and the remainder is r(x) = 9x + 10. Since deg(9x + 10) = 1 < deg(g(x)) = 2, division stops. Verification: g(x)q(x) + r(x) = (x^2 + 2x + 1)(3x - 5) + (9x + 10) = 3x^3 - 5x^2 + 6x^2 - 10x + 3x - 5 + 9x + 10 = 3x^3 + x^2 + 2x + 5 = P(x).
Solution 2: The given zeros are 2 + sqrt(3) and 2 - sqrt(3). Form the quadratic factor: (x - (2 + sqrt(3))) * (x - (2 - sqrt(3))) = ((x - 2) - sqrt(3)) * ((x - 2) + sqrt(3)) = (x - 2)^2 - 3 = x^2 - 4x + 4 - 3 = x^2 - 4x + 1. Divide P(x) by x^2 - 4x + 1: (x^4 - 6x^3 - 26x^2 + 138x - 35) / (x^2 - 4x + 1). Dividing leading terms: x^4 / x^2 = x^2. Subtracting gives -2x^3 - 27x^2 + 138x - 35. Dividing next term: -2x^3 / x^2 = -2x. Subtracting gives -35x^2 + 140x - 35. Dividing next term: -35x^2 / x^2 = -35. Subtracting gives 0. The quotient is q(x) = x^2 - 2x - 35. Factoring q(x): (x - 7)*(x + 5) = 0, giving zeros x = 7 and x = -5. Thus, the four zeros are 2 + sqrt(3), 2 - sqrt(3), 7, and -5.
Solution 3: From the Division Algorithm: P(x) = g(x) * q(x) + r(x). Rearranging to solve for g(x): g(x) * q(x) = P(x) - r(x), which means g(x) = [P(x) - r(x)] / q(x). We are given P(x) = x^3 - 3x^2 + x + 2, q(x) = x - 2, and r(x) = -2x + 4. Compute P(x) - r(x): (x^3 - 3x^2 + x + 2) - (-2x + 4) = x^3 - 3x^2 + 3x - 2. Now perform long division of (x^3 - 3x^2 + 3x - 2) by (x - 2): (x^3 / x) = x^2; subtracting (x^3 - 2x^2) leaves -x^2 + 3x - 2; (-x^2 / x) = -x; subtracting (-x^2 + 2*x) leaves x - 2; (x / x) = 1; subtracting leaves 0. Therefore, g(x) = x^2 - x + 1.