# Adding, subtracting and multiplying polynomials

Canonical: https://duckyhelper.com/learn/algebra-1/polynomials/
Updated: 2026-10-01

A polynomial is a sum of terms like \(3x^2 - 5x + 2\). To add or subtract polynomials, combine like terms, which have the same variable and the same exponent. When subtracting, change the sign of every term in the second polynomial first. To multiply, multiply every term in the first polynomial by every term in the second, then combine like terms. FOIL is that rule for two binomials.

## The key idea

**Words you will see**

| Word | Meaning | In \(3x^2 - 5x + 2\) |
| --- | --- | --- |
| term | one piece between plus and minus signs | \(3x^2\), \(-5x\), \(2\) |
| coefficient | the number in front of a variable | 3 and \(-5\) |
| like terms | same variable with the same exponent | \(3x^2\) and \(7x^2\) are like; \(3x^2\) and \(3x\) are not |
| degree | the highest exponent | 2 |
| standard form | terms written from highest exponent to lowest | already in standard form |

Multiplying uses the distributive property over and over. For two binomials, there are four products:

$$
(a + b)(c + d) = ac + ad + bc + bd
$$

Three products come up so often that they are worth memorizing:

$$
(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2 \qquad (a + b)(a - b) = a^2 - b^2
$$

## Worked examples

**Example 1: subtracting polynomials**

Problem: Simplify \((4x^2 - 3x + 7) - (x^2 + 5x - 2)\).

1. The minus sign in front of the second group changes the sign of **every** term inside it.

   $$
   (4x^2 - 3x + 7) - (x^2 + 5x - 2) = 4x^2 - 3x + 7 - x^2 - 5x + 2
   $$
2. Group like terms, then combine them.

   $$
   4x^2 - x^2 - 3x - 5x + 7 + 2 = 3x^2 - 8x + 9
   $$

Answer: \(3x^2 - 8x + 9\)

**Example 2: FOIL**

Problem: Multiply \((2x + 3)(x - 4)\).

1. First, Outer, Inner, Last: \(2x \cdot x\), \(2x \cdot (-4)\), \(3 \cdot x\), \(3 \cdot (-4)\).

   $$
   (2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12
   $$
2. Combine the two middle terms.

   $$
   2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12
   $$

Answer: \(2x^2 - 5x - 12\)

**Example 3: a binomial times a trinomial**

Problem: Multiply \((x - 2)(x^2 + 3x - 1)\).

1. FOIL only covers two binomials. Here, multiply each of the 2 terms by each of the 3 terms, so expect 6 products. First the \(x\), then the \(-2\).

   $$
   (x - 2)(x^2 + 3x - 1) = x^3 + 3x^2 - x - 2x^2 - 6x + 2
   $$
2. Combine like terms and write in standard form.

   $$
   x^3 + 3x^2 - x - 2x^2 - 6x + 2 = x^3 + x^2 - 7x + 2
   $$

Answer: \(x^3 + x^2 - 7x + 2\)

**Example 4 (test-hard): special products together**

Problem: Simplify \((3x - 5)^2 - (3x + 5)(3x - 5)\).

1. Square of a difference: \(a = 3x\) and \(b = 5\), so \(a^2 - 2ab + b^2\).

   $$
   (3x - 5)^2 = 9x^2 - 30x + 25
   $$
2. Sum times difference: \(a^2 - b^2\).

   $$
   (3x + 5)(3x - 5) = 9x^2 - 25
   $$
3. Subtract the second result, keeping it in parentheses so both of its signs change.

   $$
   9x^2 - 30x + 25 - (9x^2 - 25) = -30x + 50
   $$

Answer: \(-30x + 50\)

## Common mistakes

- **Squaring a binomial without the middle term.** \((x + 5)^2\) is \(x^2 + 10x + 25\), not \(x^2 + 25\). Fix: write it as \((x + 5)(x + 5)\) and FOIL.
- **Changing only the first sign when subtracting.** \(-(x^2 + 5x - 2)\) is \(-x^2 - 5x + 2\). All three signs change.
- **Combining unlike terms.** \(3x^2 + 2x\) stays as it is. It is not \(5x^3\) or \(5x^2\).
- **Multiplying exponents when you multiply terms.** \(x^2 \cdot x^3 = x^5\). Exponents add when you multiply.
- **Missing a product.** A 2-term times a 3-term polynomial has \(2 \times 3 = 6\) products before combining. Count them.

## Quick methods

> **Tip: The box method keeps big products organized**
>
> Draw a grid with one polynomial's terms across the top and the other's down the side. Fill each box with the product of its row and column. Then add the boxes, combining like terms, which usually sit on diagonals. It is the same math as distributing, laid out so you cannot skip a product.

> **Note: Check by plugging in a number**
>
> Pick \(x = 2\) and evaluate the problem and your answer. In Example 2, \((2 \cdot 2 + 3)(2 - 4) = -14\), and \(2(4) - 5(2) - 12 = -14\). If the numbers differ, there is a mistake. (Matching at one number is a strong hint, not a proof.)

## Practice

**5 practice questions**

1. Simplify \((5x^2 + 2x - 1) + (3x^2 - 6x + 4)\).
   A. \(8x^2 - 4x + 3\)
   B. \(8x^4 - 4x^2 + 3\)
   C. \(8x^2 + 8x + 3\)
   D. \(2x^2 - 4x + 3\)

   Answer: \(8x^2 - 4x + 3\). Add like terms: \(5x^2 + 3x^2 = 8x^2\), \(2x - 6x = -4x\), \(-1 + 4 = 3\). Adding never changes exponents, so \(8x^4\) is wrong.

2. Simplify \((3x^2 - x + 2) - (x^2 - 4x + 7)\).
   A. \(2x^2 + 3x - 5\)
   B. \(2x^2 - 5x - 5\)
   C. \(2x^2 - 5x + 9\)
   D. \(4x^2 - 5x + 9\)

   Answer: \(2x^2 + 3x - 5\). Change every sign in the second group: \(3x^2 - x + 2 - x^2 + 4x - 7\). Combine: \(2x^2 + 3x - 5\). The choices with \(-5x\) did not flip the sign of \(-4x\), and \(+9\) did not flip the sign of 7.

3. Multiply \((x + 6)(x - 2)\).
   A. \(x^2 - 12\)
   B. \(x^2 + 4x - 12\)
   C. \(x^2 - 4x - 12\)
   D. \(x^2 + 8x - 12\)

   Answer: \(x^2 + 4x - 12\). FOIL: \(x^2 - 2x + 6x - 12 = x^2 + 4x - 12\). The choice \(x^2 - 12\) multiplies only first and last terms and skips the middle.

4. Expand \((2x + 7)^2\).
   A. \(4x^2 + 49\)
   B. \(2x^2 + 28x + 49\)
   C. \(4x^2 + 14x + 49\)
   D. \(4x^2 + 28x + 49\)

   Answer: \(4x^2 + 28x + 49\). Use \(a^2 + 2ab + b^2\) with \(a = 2x\) and \(b = 7\): \(4x^2 + 2(2x)(7) + 49 = 4x^2 + 28x + 49\). The choice with \(14x\) forgot the 2 in \(2ab\), and \(4x^2 + 49\) has no middle term at all.

5. Multiply \((4x - 3)(4x + 3)\).
   A. \(16x^2 - 9\)
   B. \(16x^2 + 9\)
   C. \(16x^2 - 24x - 9\)
   D. \(8x^2 - 9\)

   Answer: \(16x^2 - 9\). This is \((a - b)(a + b) = a^2 - b^2\). The middle terms \(12x\) and \(-12x\) cancel, leaving \(16x^2 - 9\). The choice \(8x^2\) doubles \(4x\) instead of squaring it.

## Frequently asked questions

### What does FOIL mean?

FOIL stands for First, Outer, Inner, Last: the four products you get when multiplying two binomials like \((2x + 3)(x - 4)\). It is just the distributive property in a fixed order. For anything bigger than two binomials, use distribution or the box method instead.

### What are like terms?

Like terms have exactly the same variables raised to exactly the same powers. \(4x^2\) and \(-x^2\) are like terms, so they combine to \(3x^2\). \(4x^2\) and \(4x\) are not, so they stay separate. Only the coefficients change when you combine.

### What is the degree of a polynomial?

It is the highest exponent on the variable after you simplify. \(x^3 + x^2 - 7x + 2\) has degree 3. The degree tells you the general shape of the graph and, later, the most solutions an equation can have.

### How do I multiply polynomials with more than two terms?

Multiply every term in the first polynomial by every term in the second. A 2-term times a 3-term polynomial gives 6 products, and a 3-term times a 3-term gives 9. Then combine like terms. A box grid keeps track so you do not miss one.

## Related

- [How to factor polynomials](https://duckyhelper.com/learn/algebra-1/factoring/)
- [Exponent rules and how to use them](https://duckyhelper.com/learn/algebra-1/exponent-rules/)
- [Equivalent expressions on the SAT](https://duckyhelper.com/learn/sat-math/equivalent-expressions/)
- [Algebra 1 study guides](https://duckyhelper.com/learn/algebra-1/)

## Try asking Ducky

- "Why isn't (x + 5) squared just x squared plus 25?"
- "Set up a box for this multiplication with me."
- "I keep messing up the signs when I subtract polynomials. Can you check my work?"

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