Let's dive into the exciting world of polynomials and rational expressions! These mathematical concepts might seem a bit daunting at first, but trust me, once you get the hang of them, they're like building blocks for more advanced algebra. So, let's roll up our sleeves and get started!
First things first, what exactly is a polynomial? Well, it's an expression that consists of variables (usually represented by letters) and coefficients, combined using addition, subtraction, and multiplication. The term "poly" means "many," so think of polynomials as expressions with many terms.
Example
Here are some examples of polynomials:
Note
The degree of a polynomial is determined by the highest power of the variable in the expression. For instance, $3x^2 + 2x - 5$ is a second-degree polynomial (also called a quadratic).
Now that we know what polynomials are, let's look at how we can work with them.
Adding and subtracting polynomials is pretty straightforward. You simply combine like terms.
Example
Let's add $(3x^2 + 2x - 5)$ and $(2x^2 - 3x + 4)$:
$(3x^2 + 2x - 5) + (2x^2 - 3x + 4)$ $= 3x^2 + 2x - 5 + 2x^2 - 3x + 4$ $= 5x^2 - x - 1$
Tip
When subtracting polynomials, remember to distribute the negative sign to all terms in the polynomial being subtracted.
Multiplying polynomials involves using the distributive property and combining like terms.
Example
Let's multiply $(2x + 3)$ and $(x - 4)$:
$(2x + 3)(x - 4)$ $= 2x(x) + 2x(-4) + 3(x) + 3(-4)$ $= 2x^2 - 8x + 3x - 12$ $= 2x^2 - 5x - 12$
Dividing polynomials can be a bit trickier. We often use long division or synthetic division for this.
Note
For polynomial division, the dividend must have a degree greater than or equal to the divisor.
Now, let's move on to rational expressions. These are fractions where both the numerator and denominator are polynomials.
Example
Here's an example of a rational expression:
$$\frac{3x^2 + 2x - 5}{x + 2}$$
To add or subtract rational expressions, we need to find a common denominator, just like with regular fractions.
Example
Let's add $\frac{1}{x+2}$ and $\frac{2}{x-1}$:
$$\frac{1}{x+2} + \frac{2}{x-1} = \frac{1(x-1)}{(x+2)(x-1)} + \frac{2(x+2)}{(x-1)(x+2)} = \frac{x-1+2x+4}{(x+2)(x-1)} = \frac{3x+3}{x^2+x-2}$$
Multiplying rational expressions is similar to multiplying fractions. We multiply the numerators and denominators separately.
Example
Let's multiply $\frac{x+1}{x-2}$ and $\frac{x-3}{x+4}$:
$$\frac{x+1}{x-2} \cdot \frac{x-3}{x+4} = \frac{(x+1)(x-3)}{(x-2)(x+4)}$$
To divide rational expressions, we multiply by the reciprocal of the divisor.
Example
Let's divide $\frac{x^2-1}{x+3}$ by $\frac{x-1}{x+2}$:
$$\frac{x^2-1}{x+3} \div \frac{x-1}{x+2} = \frac{x^2-1}{x+3} \cdot \frac{x+2}{x-1} = \frac{(x^2-1)(x+2)}{(x+3)(x-1)}$$
Common Mistake
Don't forget to factor and simplify your rational expressions when possible. This can often make your calculations much easier!
By mastering these operations with polynomials and rational expressions, you're setting yourself up for success in more advanced algebraic concepts. Keep practicing, and soon you'll be manipulating these expressions like a pro!