A polynomial is a set of terms made up by variables, coefficients, and constants e.g.
\[ x^1 + 2 \]
\[ 2x^2 + 3x^1 + 9 \]
You’ll be asked to add, subtract, and multiply; you must understand how to complete each task.
When adding or subtracting like terms, you add the terms that share the same base and exponent.
\[ x + x = 2x \]
\[ 8y - 4y = 4y \]
Look at the following example:
\[ (3x^2 - 2x + 9) + (x^2 + 6x - 7) \]
Go down the line and identify which terms share the same base and exponent
Our first two like terms are \( 3x^2 \) and \( x^2 \). Adding these together we get \( 4x^2 \).
Next, we have our x terms (\(-2x\) and \(6x\)) that result in a sum of \(4x\).
Our constants (9 and -7) add up to 2.
Combine all your answers into one final equation
\((3x^2 - 2x + 9) + (x^2 + 6x - 7)\) = \(4x^2 + 4x + 2\).
Subtracting is the exact same thing just with a different sign.
\[ (8x^2 + 4x - 8) - (3x^2 - 2x + 12) \]
Simply subtract all the like terms from each other:
\[ 8x^2 - 3x^2 = 5x^2 \]
\[ 4x - (-2x) = 6x \]
\[ -8 - 12 = -20 \]
Our final answer is \(5x^2 + 6x - 20\).
To understand how to multiply full polynomials, it is essential you fully understand the distributive property and how to multiply numeric values with other expressions. Reference the example below:
\[ 2(x^2 + 4x + 3) \]
Since there is no operation (+ or -) between the 2 and parentheses, we know we are multiplying. Simply distribute the 2 to each individual term and multiply by its coefficient (number in front).
\[ 2(x^2) + 2(4x) + 2(3) \]
\[ 2x^2 + 8x + 6 \]
This same concept applies to multiplying variables and expressions (if you need more help with the distributive property watch this video “there will be a link here”).
Let's look at the following example:
\[ x^1 \cdot x^1 \]
To multiply variables, multiply their coefficients and add their exponents. Since the coefficient of both variables is one, the coefficient of the product will also remain one. Since we are adding their exponents and both exponents are 1, the exponent of the product will be 2. So:
\[ x^1 \cdot x^1 = x^2 \]
Let's take a look at a more complex example:
\[ (2x^2)(x^2 + 3) \]
(Note: \(3\) is equivalent to \(3x^0\))
First distribute:
\[ 2x^2(x^2) + 2x^2(3) \]
Then multiply their coefficients and add their exponents:
\[ 2x^4 + 6x^2 \]
If your still having trouble with the distributive property and multiplying polynomials, watch the video below.
For a full review of operations on polynomials