Hey there, algebra enthusiasts! Today, we're diving into the exciting world of polynomial arithmetic. Don't worry if that sounds intimidating – by the end of this study note, you'll be adding, subtracting, multiplying, and even dividing polynomials like a pro!
Before we jump into operations, let's quickly refresh our memory on what polynomials are:
A polynomial is an expression consisting of variables (usually represented by letters) and coefficients, combined using addition, subtraction, and multiplication operations. The term "poly" means "many," and "nomial" refers to "terms."
For example: $3x^2 + 2x - 5$ is a polynomial with three terms.
Note
Remember: The exponents in polynomials are always non-negative integers!
Adding and subtracting polynomials is a lot like combining like terms. The key is to identify terms with the same variables and exponents.
Example
Let's add $(2x^2 + 3x - 1)$ and $(x^2 - 2x + 4)$:
$$(2x^2 + 3x - 1) + (x^2 - 2x + 4)$$
Aligning like terms:
2x^2 + 3x - 1
+ x^2 - 2x + 4
-----------------
3x^2 + x + 3
Common Mistake
Don't forget to carry over the sign when subtracting polynomials! For example, when subtracting $(x^2 - 2x + 4)$, it becomes $(-x^2 + 2x - 4)$ in the operation.
Multiplying polynomials is where things get a bit more interesting. We'll use the distributive property to multiply each term of one polynomial by every term of the other.
Example
Let's multiply $(x + 2)$ by $(x - 3)$:
$$(x + 2)(x - 3)$$
Distributing:
x(x) + x(-3) + 2(x) + 2(-3)
= x^2 - 3x + 2x - 6
= x^2 - x - 6
Tip
When multiplying polynomials with multiple terms, the FOIL method (First, Outer, Inner, Last) can be helpful for binomials. For larger polynomials, creating a grid can help organize your work!
Division of polynomials is often the trickiest operation, but with practice, it becomes second nature. We'll focus on two methods: long division and synthetic division.
This method is similar to long division with numbers, but we work with terms instead of digits.
Example
Let's divide $x^2 + 5x + 6$ by $x + 2$:
x + 3
___________
x + 2 ) x^2 + 5x + 6
x^2 + 2x
___________
3x + 6
3x + 6
___________
0
So, $(x^2 + 5x + 6) ÷ (x + 2) = x + 3$ with no remainder.
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form $(x - a)$.
Example
Let's divide $x^2 - 7x + 12$ by $x - 3$:
3 | 1 -7 12
| 3 -12
_______________
1 -4 0
The result is $x - 4$ with no remainder.
Note
Synthetic division works when dividing by $(x - a)$. If you need to divide by $(x + a)$, just use $-a$ in your synthetic division setup!
And there you have it! You've just taken a whirlwind tour through the world of polynomial arithmetic. Remember, practice makes perfect, so don't be afraid to work through lots of examples. Before you know it, you'll be manipulating polynomials with ease!
Tip
Always check your work by performing the inverse operation. For example, after dividing, multiply your quotient by the divisor and add the remainder (if any) – you should get back your original polynomial!
Keep up the great work, and happy polynomial crunching!