Subtracting polynomials may sound complicated, but it’s quite manageable if you follow a systematic approach. Let’s break it down step by step.
- Arrange the Polynomials
First, write down the polynomials you want to subtract in a standard form, which means arranging the terms in descending order of their degrees. For example, if you have two polynomials $P(x) = 3x^3 + 2x^2 – x + 5$ and $Q(x) = x^3 – 4x^2 + 2x – 3$, arrange them as shown.
- Distribute the Negative Sign
When subtracting polynomials, you need to distribute the negative sign across each term of the polynomial being subtracted. This means changing the sign of every term in the second polynomial. For example:$(3x^3 + 2x^2 – x + 5) – (x^3 – 4x^2 + 2x – 3)$
becomes:
$3x^3 + 2x^2 – x + 5 – x^3 + 4x^2 – 2x + 3$
- Combine Like Terms
Next, combine like terms. Like terms are terms that have the same variable raised to the same power. So, group the $x^3$ terms, $x^2$ terms, $x$ terms, and constant terms together:$3x^3 – x^3 + 2x^2 + 4x^2 – x – 2x + 5 + 3$
- Simplify Each Group
Now, simplify each group by performing the arithmetic operations. For our example:- Combine the $x^3$ terms: $3x^3 – x^3 = 2x^3$
- Combine the $x^2$ terms: $2x^2 + 4x^2 = 6x^2$
- Combine the $x$ terms: $-x – 2x = -3x$
- Combine the constant terms: $5 + 3 = 8$
- Write the Result
Finally, write down the simplified polynomial:$2x^3 + 6x^2 – 3x + 8$
Example Problem
Let’s go through another example to solidify the steps. Suppose we want to subtract $R(x) = 5x^2 – 3x + 7$ from $S(x) = 2x^2 + x – 4$
- Arrange the polynomials:
$S(x) = 2x^2 + x – 4$
$R(x) = 5x^2 – 3x + 7$
- Distribute the negative sign:
$2x^2 + x – 4 – (5x^2 – 3x + 7)$
$= 2x^2 + x – 4 – 5x^2 + 3x – 7$
- Combine like terms:
$2x^2 – 5x^2 + x + 3x – 4 – 7$
- Simplify each group:
- Combine the $x^2$ terms: $2x^2 – 5x^2 = -3x^2$
- Combine the $x$ terms: $x + 3x = 4x$
- Combine the constant terms: $-4 – 7 = -11$
- Write the result:
$-3x^2 + 4x – 11$
Conclusion
By following these steps—arranging the polynomials, distributing the negative sign, combining like terms, and simplifying—you can easily subtract any polynomials. Practice a few problems on your own to get the hang of it. Happy calculating!