These lessons, with videos, examples and step-by-step solutions, help High School students learn how to rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
Related Pages
Polynomial Long Division
Synthetic Division
Common Core Algebra
Common Core Mathematics
Rewrite rational expressions,
a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) by using inspection, long division, or synthetic division and comparing coefficients.
Use a computer algebra system for complicated examples to assist with building a broader conceptual understanding.
Common Core: HSA-APR.D.6
Polynomial Division using a computer algebra system (for example Wolfram Alpha)
Example:
Divide x^{3} + x^{2} − 7 by x − 3 by comparing coefficients.
Rewrite the expression
x^{3} + x^{2} − 7 ≡ (Ax^{2} + Bx + C)(x − 3) + D
Expanding the right side we get,
(Ax^{2} + Bx + C)(x − 3) + D = Ax^{3} + (−3A + B)x^{2} + (−3B + C)x − 3C + D
Matching coefficients of the various powers of x on the left and right hand sides, we get
x^{3} + x^{2} − 7 ≡ Ax^{3} + (−3A + B)x^{2} + (−3B + C)x − 3C + D
A = 1
(−3A + B) = 1 ⇒ B = 4
(−3B + C) = 0 ⇒ C =12
− 3C + D = -7 ⇒ D = 29
The expression can then be written as
x^{3} + x^{2} − 7 ≡ (x^{2} + 4x + 12)(x − 3) + 29
Thus, (x^{3} + x^{2} − 7)/(x − 3) = x^{2} + 4x + 12 + 29/(x − 3)
Polynomial Division & Equating Coefficients
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