algebra911.comAdding And Subtracting Polynomials
Ever looked at a long string of variables and exponents and felt lost? You're in the right place! This guide breaks down how to add and subtract polynomials into simple, manageable steps. We'll master the core concept of 'like terms' and tackle any problem with confidence.
What Are Polynomials, Anyway?
A polynomial is an algebraic expression made up of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. The key rule is that the exponents on the variables must be non-negative whole numbers (\(0, 1, 2, 3, ...)\). You won't see variables in the denominator of a fraction or with negative exponents in a polynomial.
Polynomials are built from smaller parts called terms. A term is a single piece of the polynomial, like \(5x^2\) or \(-3y\) or \(7\). We often classify polynomials by their number of terms:
- Monomial: An expression with one term (e.g., \(12a^3\)).
- Binomial: An expression with two terms (e.g., \(x^2 - 9\)).
- Trinomial: An expression with three terms (e.g., \(4x^2 + 8x + 3\)).
Any expression with more than three terms is usually just called a polynomial. Each term has a coefficient (the number part) and a variable part (the letter part with its exponent).
| Expression | Number of Terms | Classification | Terms | Coefficients |
|---|---|---|---|---|
| \(8x^4\) | 1 | Monomial | \(8x^4\) | \(8\) |
| \(5y^2 - 11\) | 2 | Binomial | \(5y^2\), \(-11\) | \(5, -11\) |
| \(a^2 - 6a + 9\) | 3 | Trinomial | \(a^2\), \(-6a\), \(9\) | \(1, -6, 9\) |
| \(2x^3 + x^2 - 7x + 4\) | 4 | Polynomial | \(2x^3\), \(x^2\), \(-7x\), \(4\) | \(2, 1, -7, 4\) |
What's the Secret to Adding and Subtracting Polynomials?
The entire process of adding and subtracting polynomials hinges on one fundamental concept: combining like terms. If you can master this, you're 90% of the way there. So, what are like terms?
Like terms are terms that have the exact same variables raised to the exact same exponents. The coefficients can be different; that's perfectly fine. Think of it like sorting fruit. You can add apples to other apples, and bananas to other bananas, but you can't combine an apple and a banana into a 'banapple'.
- \(7x^2\) and \(-3x^2\) are like terms. They both have the variable \(x\) raised to the power of \(2\).
- \(5y\) and \(10y\) are like terms. They both have the variable \(y\) raised to the power of \(1\).
- \(-ab^3\) and \(4ab^3\) are like terms. They both have \(a\) to the power of \(1\) and \(b\) to the power of \(3\).
Conversely, here are examples of terms that are NOT like terms:
- \(4x^3\) and \(4x^2\) are not like terms because the exponents are different (\(3\) vs \(2\)).
- \(6a\) and \(6b\) are not like terms because the variables are different (\(a\) vs \(b\)).
- \(2x\) and \(8\) are not like terms because one has a variable and the other is a constant.
When you add or subtract polynomials, you are simply finding the like terms and combining their coefficients.
How Do You Add Polynomials?
Adding polynomials is a straightforward process of combining the like terms we just discussed. The parentheses around polynomials are often just there for clarity, and when you're adding, you can usually just drop them. There are two common methods for organizing your work: the horizontal method and the vertical method.
The Horizontal Method
This method involves writing the problem on a single line and then regrouping the like terms together before adding.
- Remove the parentheses. Since it's addition, the signs don't change.
- Identify and group like terms. You can use different colors, underlines, or shapes to keep track.
- Add the coefficients of the like terms. The variable part (including the exponent) stays the same.
The Vertical Method
This method looks very similar to how you learned to add large numbers in elementary school. It's great for keeping your work neat, especially with long polynomials.
- Write the first polynomial. It's helpful to write it in standard form (highest exponent to lowest).
- Write the second polynomial directly below it. Carefully line up the like terms in columns. If a term is missing, leave a blank space.
- Add down each column, just like regular addition.
Both methods will give you the exact same answer, so you can choose the one you find easier to use!
Let's Walk Through an Addition Example
Let's put the theory into practice. We'll solve the same problem using both methods to show they produce the same result.
Find the sum of \( (4x^3 + 2x^2 - 5x + 8) \) and \( (x^3 - 3x^2 - 2) \).
Method 1: Horizontal
- Remove parentheses:
\( 4x^3 + 2x^2 - 5x + 8 + x^3 - 3x^2 - 2 \) - Group like terms:
We can rearrange the expression to put like terms next to each other. Pay close attention to the signs.
\( (4x^3 + x^3) + (2x^2 - 3x^2) + (-5x) + (8 - 2) \) - Combine the coefficients:
For the \(x^3\) terms: \(4 + 1 = 5\). So we have \(5x^3\).
For the \(x^2\) terms: \(2 - 3 = -1\). So we have \(-x^2\).
The \(-5x\) term has no like term to combine with.
For the constants: \(8 - 2 = 6\). - Final Answer:
\( 5x^3 - x^2 - 5x + 6 \)
Method 2: Vertical
- Line up the polynomials. Notice we leave a gap where the first polynomial has an \(x\) term but the second does not. We also add a \(0\) placeholder for clarity.
- Add each column:
- Final Answer:
\( 5x^3 - x^2 - 5x + 6 \)
As you can see, both methods yield the same correct answer.
How Is Subtracting Polynomials Different?
Subtraction adds one crucial extra step right at the beginning. This is the step where most mistakes happen, so pay close attention! When you subtract a polynomial, you are subtracting its entirety, not just the first term.
The key is to distribute the negative sign to every single term in the polynomial being subtracted. This means you will flip the sign of each term inside the second set of parentheses. After you do that, the problem turns into an addition problem, and you can proceed exactly as before.
For example, let's look at \( (5x + 4) - (2x + 1) \).
You can't just do \(5x - 2x\) and \(4 + 1\). That's wrong. You must distribute the negative:
\( (5x + 4) - (2x + 1) \) becomes \( 5x + 4 - 2x - 1 \).
Notice that the \(+2x\) became \(-2x\) and the \(+1\) became \(-1\). Now you can combine like terms as usual: \((5x - 2x) + (4 - 1) = 3x + 3\). This is the most important rule in polynomial subtraction: first distribute, then add.
Let's Tackle a Subtraction Problem
Let's work through a full subtraction example. We'll focus on the horizontal method here, as it clearly shows the distribution step.
Subtract \( (3y^2 - 8y + 5) \) from \( (10y^2 + 2y - 7) \).
First, write the problem correctly. 'Subtract A from B' means \(B - A\).
- Distribute the negative sign. This is the critical first step. The negative in front of the second parenthesis applies to every term inside it. Think of it as multiplying by \(-1\).
\( - (3y^2 - 8y + 5) = -3y^2 + 8y - 5 \)
Our expression now becomes an addition problem:
\( 10y^2 + 2y - 7 - 3y^2 + 8y - 5 \) - Group like terms. Now we proceed just like we did with addition.
\( (10y^2 - 3y^2) + (2y + 8y) + (-7 - 5) \) - Combine the coefficients.
For the \(y^2\) terms: \(10 - 3 = 7\). So we have \(7y^2\).
For the \(y\) terms: \(2 + 8 = 10\). So we have \(10y\).
For the constants: \(-7 - 5 = -12\). - Final Answer:
\( 7y^2 + 10y - 12 \)
Remember, the most common error is only changing the sign of the first term (\(3y^2\)) and forgetting the others. Distribute the negative to every term!
Can You Add and Subtract in the Same Problem?
Absolutely. In algebra, you'll often see problems that involve multiple operations. The key is to handle them step-by-step, following the order of operations. When dealing with adding and subtracting polynomials, work from left to right, being careful to distribute any negatives before you start combining terms.
Simplify the expression: \( (8a^3 - 2a^2 + 5) - (a^3 - 3a) + (4a^2 + 6a - 1) \).
- Handle the subtraction first by distributing the negative. The only subtraction is on the second polynomial. The plus sign in front of the third polynomial doesn't change anything, so we can just drop its parentheses.
The expression becomes:
\( 8a^3 - 2a^2 + 5 - a^3 + 3a + 4a^2 + 6a - 1 \) - Identify and group all like terms from the entire expression. Let's look for all the \(a^3\) terms, then all the \(a^2\) terms, and so on.
\( (8a^3 - a^3) + (-2a^2 + 4a^2) + (3a + 6a) + (5 - 1) \) - Combine the coefficients for each group.
\(a^3\) terms: \(8 - 1 = 7\) \( \(ightarrow 7a^3 \)
\(a^2\) terms: \(-2 + 4 = 2\) \ightarrow 2a^2 \)
\(a\) terms: \(3 + 6 = 9\) \( \(ightarrow 9a \)
Constants: \(5 - 1 = 4\) \ightarrow 4 \) - Write the final simplified polynomial.
\( 7a^3 + 2a^2 + 9a + 4 \)
What Are the Common Mistakes to Avoid?
Working with polynomials is generally straightforward, but a few common pitfalls can trip students up. Being aware of them is the best way to avoid making them yourself.
- Forgetting to Distribute the Negative: This is by far the most common mistake in subtraction. Students often change the sign of the first term in the parentheses but forget the rest. Always distribute the negative to every single term being subtracted.
- Combining Unlike Terms: You cannot add \(3x^2\) and \(5x\) to get \(8x^3\). They are not like terms because their exponents are different. The simplest they can be written is just \(3x^2 + 5x\). Don't combine apples and oranges!
- Changing Exponents: When you add or subtract like terms, the variable and its exponent do not change. For example, \(4x^2 + 5x^2 = 9x^2\), NOT \(9x^4\). You are simply counting how many \(x^2\) you have in total.
- Sign Errors with Coefficients: Simple arithmetic mistakes can still happen. Be careful when adding and subtracting negative numbers, like \(-8 - (-3)\). Remember that subtracting a negative is the same as adding a positive, so \(-8 - (-3) = -8 + 3 = -5\).
Quick Summary for Review
Need a quick refresher? Here are the essential steps for adding and subtracting polynomials.
To Add Polynomials
- Drop Parentheses: Remove the parentheses from the expression.
- Group Like Terms: Identify terms with the same variable and exponent.
- Combine Coefficients: Add the numbers in front of the like terms. The variable and exponent stay the same.
To Subtract Polynomials
- Distribute the Negative: Change the sign of every term in the polynomial being subtracted.
- Change to Addition: Rewrite the problem as an addition problem using the new signs.
- Follow Addition Rules: Proceed by grouping like terms and combining their coefficients.
Frequently Asked Questions
What's the difference between a monomial, binomial, and trinomial?
It's all about the number of terms in the expression. A monomial has just one term (like \(5x^2\)), a binomial has two terms (like \(3x - 1\)), and a trinomial has three terms (like \(x^2 + 4x + 4\)).
Can you add terms with different variables, like x and y?
No, you cannot combine them into a single term because they are not 'like terms'. The expression \(4x + 2y\) is as simple as it gets. You can only combine terms that have the exact same variable parts.
What happens to the exponents when you add or subtract polynomials?
Nothing at all! The exponents on the variables stay exactly the same. When you combine like terms, such as \(5x^2 + 2x^2\), you only add the coefficients to get \(7x^2\). The exponent doesn't change.
Why do you have to distribute the negative sign when subtracting?
Because you are subtracting the entire polynomial quantity, not just its first term. Think of the negative sign as a \(-1\) that needs to be multiplied by every term inside the parentheses, following the distributive property.
Does the order matter when you add polynomials?
For addition, the order does not matter; it's commutative, so \(A + B\) is the same as \(B + A\). However, for subtraction, order is extremely important! \(A - B\) is very different from \(B - A\).
What is the 'degree' of a polynomial?
The degree of a polynomial is the value of the highest exponent on the variable in any single term. For example, the degree of \(6x^5 - 2x^3 + 8\) is \(5\) because that is the largest exponent.
Is an expression with a negative exponent like \(4x^{-2} + 3\) a polynomial?
No, it is not. A strict rule for an expression to be a polynomial is that all variables must have non-negative, whole number exponents. Since this expression has an exponent of \(-2\), it is not a polynomial.