Polynomial Addition

Grade 9 · algebra · 82 practice problems · read aloud

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Polynomial Addition

🧠 What is Polynomial Addition?

Polynomial addition is the process of combining two or more polynomials by adding their like terms. A polynomial is an expression with variables and coefficients, like 3x² + 2x - 5. We use this skill to simplify expressions in algebra, which is essential for solving equations and graphing functions.

📝 How to Add Polynomials: A Step-by-Step Guide

  1. Step 1: Rewrite the Expression
    Write the polynomials next to each other, usually in parentheses with a plus sign.
  2. Step 2: Identify Like Terms
    Like terms have the exact same variable and exponent. Group them together. Constants (numbers without variables) are also like terms.
  3. Step 3: Combine Coefficients
    Add or subtract the coefficients (the numbers in front) of the like terms.
  4. Step 4: Write the Final Answer
    Write the new polynomial in standard form, usually from highest to lowest degree.

🔍 Worked Examples

Example 1: Add (2x² + 4x + 3) + (5x² - 2x - 1)

  1. Rewrite: 2x² + 4x + 3 + 5x² - 2x - 1
  2. Group Like Terms: (2x² + 5x²) + (4x - 2x) + (3 - 1)
  3. Combine: 7x² + 2x + 2

Answer: 7x² + 2x + 2

Example 2: Add (3y³ - y + 7) + (y³ + 4y - 2)

  1. Rewrite: 3y³ - y + 7 + y³ + 4y - 2
  2. Group Like Terms: (3y³ + y³) + (-y + 4y) + (7 - 2)
  3. Combine: 4y³ + 3y + 5

Answer: 4y³ + 3y + 5

⚠️ Common Mistakes to Avoid

  • Adding Exponents: When adding x² + x², you get 2x², NOT x⁴. You only add the coefficients, not the exponents.
  • Ignoring Signs: Pay close attention to positive and negative signs, especially when a term is being subtracted.
  • Combining Unlike Terms: You cannot combine 3x and 2x². They are not like terms!

💡 Tips & Tricks

  • Highlighting: Use different colored highlighters or pens to mark each group of like terms.
  • Vertical Method: For complex polynomials, line them up vertically by their degree (like stacking numbers for addition).
  • Memory Aid: Remember "CLT" - Combine Like Terms!

🎯 How to Practice

To master polynomial addition:

  • Start with simple binomials (two terms) and work up to polynomials with more terms.
  • Create your own practice problems and check your work.
  • Use online math practice websites for instant feedback.
  • Practice with a partner—one person writes a problem, the other solves it.

Practice problems

6 of the 82, worked through step by step — try them before opening the answer.

1 (3x² - 2x + 5) + (2x² + 4x - 3) = ?

Hint: Combine like terms by adding coefficients of terms with the same variable and exponent.

Show the answer

Answer: 5x² + 2x + 2

  1. Remove the parentheses since we are adding. This gives: 3x² - 2x + 5 + 2x² + 4x - 3.
  2. Identify like terms. Like terms are terms with the same variable and exponent. - x² terms: 3x² and 2x² - x terms: -2x and 4x - constant terms: 5 and -3
  3. Combine the x² terms. 3x² + 2x² = (3 + 2)x² = 5x².
  4. Combine the x terms. -2x + 4x = (-2 + 4)x = 2x.
  5. Combine the constant terms. 5 + (-3) = 5 - 3 = 2.
  6. Write the final simplified polynomial. Putting it all together: 5x² + 2x + 2. Final answer: 5x² + 2x + 2

We are adding two polynomials: (3x² - 2x + 5) + (2x² + 4x - 3).

2 (3x² - 5x + 7) + (2x² + 4x - 3) = ?

Hint: Combine like terms by adding coefficients of terms with the same variable and exponent.

Show the answer

Answer: 5x² - x + 4

  1. Remove parentheses** Since it's addition, the parentheses don't change the signs: 3x² - 5x + 7 + 2x² + 4x - 3 **
  2. Group like terms** - x² terms: 3x² + 2x² - x terms: -5x + 4x - constant terms: 7 - 3 **
  3. Combine x² terms** 3x² + 2x² = (3 + 2)x² = 5x² **
  4. Combine x terms** -5x + 4x = (-5 + 4)x = -1x, which is -x **
  5. Combine constants** 7 - 3 = 4 **
  6. Write the final polynomial** 5x² - x + 4 **Final Answer:** 5x² - x + 4

Let's add the two polynomials step by step. We have: (3x² - 5x + 7) + (2x² + 4x - 3) **

3 (3x² - 5x + 7) + (2x² + 4x - 9) = ?

Hint: Combine like terms by adding coefficients of terms with the same variable and exponent.

Show the answer

Answer: 5x² - x - 2

  1. Remove parentheses** Since it's addition, the parentheses don't change the signs: 3x² - 5x + 7 + 2x² + 4x - 9 --- **
  2. Group like terms** - x² terms: 3x² + 2x² - x terms: -5x + 4x - constant terms: 7 - 9 --- **
  3. Combine x² terms** 3x² + 2x² = (3 + 2)x² = 5x² --- **
  4. Combine x terms** -5x + 4x = (-5 + 4)x = -1x, which is written as -x --- **
  5. Combine constant terms** 7 - 9 = -2 --- **
  6. Write the final expression** 5x² - x - 2 --- **Final Answer:** 5x² - x - 2

Let's add the two polynomials step by step. We have: (3x² - 5x + 7) + (2x² + 4x - 9) --- **

4 (3x² - 2x + 5) + (2x² + 4x - 3) - (x² - 3x + 1) = ?

Hint: Combine like terms by grouping coefficients of the same degree, remembering to distribute negative signs when subtracting polynomials.

Show the answer

Answer: 4x² + 5x + 1

  1. Remove parentheses carefully** The first two parentheses are added, so they stay the same: 3x² - 2x + 5 + 2x² + 4x - 3 The third parenthesis is subtracted, so we reverse all signs inside it: - (x² - 3x + 1) = -x² + 3x - 1 Now the whole expression is: 3x² - 2x + 5 + 2x² + 4x - 3 - x² + 3x - 1 --- **
  2. Combine like terms for x² terms** 3x² + 2x² - x² = (3 + 2 - 1)x² = 4x² --- **
  3. Combine like terms for x terms** -2x + 4x + 3x = (-2 + 4 + 3)x = 5x --- **
  4. Combine constant terms** 5 - 3 - 1 = (5 - 3) - 1 = 2 - 1 = 1 --- **
  5. Write the final expression** 4x² + 5x + 1 --- **Final answer:** 4x² + 5x + 1

Let's solve step by step. We have: (3x² - 2x + 5) + (2x² + 4x - 3) - (x² - 3x + 1) --- **

5 (3x² - 5x + 7) + (2x² + 4x - 3) - (x² - 2x + 1) = ?

Hint: Combine like terms by grouping coefficients of the same degree, paying attention to sign changes when subtracting polynomials.

Show the answer

Answer: 4x² + x + 3

  1. Remove parentheses carefully, especially the minus sign before the third group.** First group: 3x² - 5x + 7 Second group: + 2x² + 4x - 3 Third group: - (x² - 2x + 1) = -x² + 2x - 1 So the expression becomes: 3x² - 5x + 7 + 2x² + 4x - 3 - x² + 2x - 1 --- **
  2. Combine like terms (x² terms, x terms, constants).** x² terms: 3x² + 2x² - x² = (3 + 2 - 1)x² = 4x² x terms: -5x + 4x + 2x = (-5 + 4 + 2)x = 1x = x Constants: 7 - 3 - 1 = (7 - 3) - 1 = 4 - 1 = 3 --- **
  3. Write the final simplified expression.** 4x² + x + 3 --- **Final Answer:** 4x² + x + 3

Let's solve step by step. We have: (3x² - 5x + 7) + (2x² + 4x - 3) - (x² - 2x + 1) --- **

6 (3x² - 5x + 7) + (2x² + 4x - 9) - (x² - 3x + 2) = ?

Hint: Combine like terms by adding coefficients of terms with the same variable and exponent, remembering to distribute negative signs when subtracting polynomials.

Show the answer

Answer: 4x² + 2x - 4

  1. Remove parentheses carefully** The first two parentheses are added, so they stay as they are: 3x² - 5x + 7 + 2x² + 4x - 9 The third parenthesis is subtracted, so we reverse all its signs: - (x² - 3x + 2) = -x² + 3x - 2 So the whole expression becomes: 3x² - 5x + 7 + 2x² + 4x - 9 - x² + 3x - 2 --- **
  2. Combine like terms (x² terms)** 3x² + 2x² - x² = (3 + 2 - 1)x² = 4x² --- **
  3. Combine x terms** -5x + 4x + 3x = (-5 + 4 + 3)x = 2x --- **
  4. Combine constant terms** 7 - 9 - 2 = (7 - 9) - 2 = (-2) - 2 = -4 --- **
  5. Write final expression** 4x² + 2x - 4 --- **Final answer:** 4x² + 2x - 4

Let's solve step by step. We have: (3x² - 5x + 7) + (2x² + 4x - 9) - (x² - 3x + 2) --- **

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