What is Polynomial Subtraction?

Polynomial subtraction is a fundamental concept in algebra, where we subtract one polynomial from another. A polynomial is an expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. For example, $3x^2 + 2x – 5$ is a polynomial.

Steps to Subtract Polynomials

1. Arrange Like Terms

First, write both polynomials in standard form, aligning like terms (terms with the same exponent). For instance, if you have $3x^2 + 2x – 5$ and $x^2 – 3x + 4$, align them as follows:

$begin{array}{r}
3x^2 + 2x – 5

• (x^2 – 3x + 4)
  end{array}$

2. Distribute the Negative Sign

Subtracting a polynomial is the same as adding its opposite. Distribute the negative sign to each term of the polynomial being subtracted:

$begin{array}{r}
3x^2 + 2x – 5

• x^2 + 3x – 4
  end{array}$

3. Combine Like Terms

Combine the like terms by performing the subtraction for each corresponding pair of terms:

$begin{array}{r}
(3x^2 – x^2) + (2x + 3x) + (-5 – 4)
2x^2 + 5x – 9
end{array}$

Thus, the result of subtracting $x^2 – 3x + 4$ from $3x^2 + 2x – 5$ is $2x^2 + 5x – 9$

Example Problem

Let’s work through another example to solidify our understanding. Suppose we want to subtract $2x^3 – 4x + 6$ from $5x^3 + x^2 – 3x + 1$

1. Arrange Like Terms
   Write the polynomials, aligning like terms:

   $begin{array}{r}
   5x^3 + x^2 – 3x + 1

   • (2x^3 + 0x^2 – 4x + 6)
     end{array}$

2. Distribute the Negative Sign
   Distribute the negative sign to each term of the polynomial being subtracted:

   $begin{array}{r}
   5x^3 + x^2 – 3x + 1

   • 2x^3 – 0x^2 + 4x – 6
     end{array}$

3. Combine Like Terms
   Combine the like terms:

   $begin{array}{r}
   (5x^3 – 2x^3) + (x^2 – 0x^2) + (-3x + 4x) + (1 – 6)
   3x^3 + x^2 + x – 5
   end{array}$

   Therefore, the result of subtracting $2x^3 – 4x + 6$ from $5x^3 + x^2 – 3x + 1$ is $3x^3 + x^2 + x – 5$

Conclusion

Polynomial subtraction involves aligning like terms, distributing the negative sign, and combining like terms. Mastering this process is essential for more complex algebraic operations and solving polynomial equations. With practice, polynomial subtraction becomes a straightforward and manageable task.

Citations

1. 1. Khan Academy – Polynomial Subtraction
2. 2. Math is Fun – Polynomials
3. 3. Purplemath – Adding and Subtracting Polynomials