Adding and Subtracting Polynomials

By the end of this lesson, you will be able to add and subtract polynomials, handle the minus sign in front of a bracket without losing terms, and write the answer in standard form.

It Is Combining Like Terms Again

You already know how to do this. Adding polynomials is the combining-like-terms lesson from the first unit, applied to longer expressions.

$$(3x^2 + 2x - 1) + (x^2 - 5x + 4)$$

Drop the brackets, because everything is being added, and gather:

$$3x^2 + x^2 + 2x - 5x - 1 + 4 = 4x^2 - 3x + 3$$

Only matching powers combine. $x^2$ terms with $x^2$ terms, $x$ with $x$, constants with constants. $3x^2$ and $2x$ stay apart forever, exactly as apples and oranges did, and the degree is what decides the kind of thing a term is.

Adding never changes the degree beyond the higher of the two, and often the answer has fewer terms than you started with, which is the whole point of doing it.

Subtraction: The Bracket Is the Lesson

$$(5x^2 + 3x - 2) - (2x^2 - 4x + 7)$$

The minus applies to every term inside the second bracket, not just the first. Distribute it before combining anything:

$$5x^2 + 3x - 2 - 2x^2 + 4x - 7$$

Notice all three signs in the second bracket flipped: $-2x^2$ stayed negative after being subtracted, $-4x$ became $+4x$, and $+7$ became $-7$.

$$3x^2 + 7x - 9$$

This single step accounts for more lost marks than everything else in the lesson combined. The reliable habit is to write the flipped bracket out as its own line before gathering, rather than doing both at once.

$$-(2x^2 - 4x + 7) = -2x^2 + 4x - 7$$

Example 1: Addition

Here is the whole method on one, worked all the way through. $$(3x^2 + 2x - 1) + (x^2 - 5x + 4)$$

• Drop the brackets and write every term in a row 3x^2 + 2x - 1 + x^2 - 5x + 4
• Combine the x-squared terms, carrying the rest across 4x^2 + 2x - 1 - 5x + 4
• Combine the x terms 4x^2 - 3x - 1 + 4
• Combine the constants 4x^2 - 3x + 3

Worked Example 1: Your Turn on Addition

$$(2x^3 - x + 6) + (4x^3 + 5x^2 - 2)$$

Worked Example 2: Subtraction, One Step at a Time

$$(5x^2 + 3x - 2) - (2x^2 - 4x + 7)$$

Sort It: What Does the Minus Do

Each expression is the second bracket of a subtraction. What does it become?

Worked Example 3: A Missing Power

Polynomials often skip a power, and lining terms up is where that bites. $$(4x^3 + 2x - 5) - (x^3 - 3x^2 + 2x)$$

Common Mistakes

Mistake 1: Distributing the minus to the first term only

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