Adding and Subtracting Polynomials

Quick Reference

Operation	Rule	Key step
Add polynomials	Combine like terms	$(7 x^{2} + 3 x) + (2 x^{2} - x) = 9 x^{2} + 2 x$
Subtract polynomials	Distribute the $-$ , then combine like terms	$(7 x^{2} + 3 x) - (2 x^{2} - x) = 5 x^{2} + 4 x$

The Rule

To add or subtract polynomials, add or subtract the coefficients of like terms.

Recall that like terms are terms with the same variable raised to the same exponent. Only like terms can be combined; terms with different exponents are left as separate terms in the result.

Adding Polynomials

To add two polynomials, remove the parentheses and collect like terms. The parentheses around the first polynomial can always be dropped without changing any signs. The parentheses around the second polynomial can also be dropped when adding, since a positive sign in front of parentheses does not change any signs inside.

Example 1. Add $(7 x^{3} - 6 x^{2} + 4 x + 9) + (x^{3} + 4 x^{2} - 2)$ .

Solution

\begin{aligned} (7x^{3}-6x^{2}+4x+9)+(x^{3}+4x^{2}-2) &= (7+1)x^{3}+(-6+4)x^{2}+(4+0)x+(9-2) \\ &= 8x^{3}-2x^{2}+4x+7 \end{aligned}

Subtracting Polynomials

Subtraction requires one extra step: the minus sign in front of the parentheses must be distributed to every term inside before combining like terms. A minus sign in front of a parenthesis changes the sign of every term inside.

Key rule for subtraction: $- (a + b) = - a - b$ and $- (a - b) = - a + b$ .
Distribute the negative sign to all terms, not just the first one.

Example 2. Subtract $(- 3 x^{4} - 8 x) - (5 x^{3} - 3 x + \sqrt{2})$ .

Solution

Distribute the minus sign to every term in the second polynomial:

(- 3 x^{4} - 8 x) - (5 x^{3} - 3 x + \sqrt{2}) = - 3 x^{4} - 8 x - 5 x^{3} + 3 x - \sqrt{2} .

Then collect like terms: \begin{aligned} &= (-3-0)x^{4}+(0-5)x^{3}+(-8+3)x+(0-\sqrt{2}) \\ &= -3x^{4}-5x^{3}-5x-\sqrt{2} \end{aligned}

Vertical Method

For longer polynomials it can be helpful to write one polynomial directly above the other, aligning like terms in columns, and then add or subtract column by column. This is called the vertical method.

If one polynomial is missing a term that the other has, write $0$ as a placeholder for that term so that the columns stay aligned.

Example 3. Subtract $(5 x^{3} - 3 x + \sqrt{2})$ from $(- 3 x^{4} - 8 x)$ using the vertical method.

Solution

Write the first polynomial on top, inserting $0$ for missing degrees:

\begin{array}{r} - 3 x^{4} + 0 x^{3} - 8 x + 0 \\ - (5 x^{3} - 3 x + \sqrt{2}) \end{array}

Change the sign of every term in the bottom row and add:

\begin{array}{r} - 3 x^{4} + 0 x^{3} - 8 x + 0 \\ + 0 x^{4} - 5 x^{3} + 3 x - \sqrt{2} \\ - 3 x^{4} - 5 x^{3} - 5 x - \sqrt{2} \end{array}

Further Examples

Example 4. Simplify $(2 x^{3} - 5 x + 1) + (x^{2} + 3 x - 4) - (x^{3} + x^{2} - 2)$ .

Solution

The first two polynomials are added and the third is subtracted. Distribute the minus sign to every term of the third polynomial: \begin{aligned} (2x^{3}&-5x+1)+(x^{2}+3x-4)-(x^{3}+x^{2}-2)\\ &= 2x^{3}-5x+1+x^{2}+3x-4-x^{3}-x^{2}+2. \end{aligned} Now collect like terms by degree: \begin{aligned} &= (2-1)x^{3}+(1-1)x^{2}+(-5+3)x+(1-4+2) \\ &= x^{3}+0\cdot x^{2}-2x-1 \\ &= x^{3}-2x-1 \end{aligned} Note that the first and third polynomials both have degree 3, while the second has degree 2. The

x^{2}

terms cancel completely, so the result has no

x^{2}

term.

Example 5. Let $P = 4 x^{2} - 3 x + 2$ and $Q = x^{2} + 2 x - 1$ . Find $P - 3 Q$ .

Solution

First multiply

Q

by $3$, distributing the scalar to every term:

3 Q = 3 (x^{2} + 2 x - 1) = 3 x^{2} + 6 x - 3.

Then subtract:

P - 3 Q = (4 x^{2} - 3 x + 2) - (3 x^{2} + 6 x - 3) .

Distribute the minus sign and collect like terms: \begin{aligned} &= 4x^{2}-3x+2-3x^{2}-6x+3 \\ &= (4-3)x^{2}+(-3-6)x+(2+3) \\ &= x^{2}-9x+5 \end{aligned}

Common Mistakes

Forgetting to distribute the minus sign to every term. The most frequent error in polynomial subtraction is changing the sign of only the first term inside the parentheses. For example:

(x^{2} + 3 x) - (2 x^{2} - x) \neq x^{2} + 3 x - 2 x^{2} - x .

The correct step is:

(x^{2} + 3 x) - (2 x^{2} - x) = x^{2} + 3 x - 2 x^{2} + x = - x^{2} + 4 x .

Combining unlike terms. Only terms with the same variable and the same exponent may be combined. For example, $3 x^{2}$ and $3 x$ are unlike terms and cannot be added to give $6 x^{2}$ or $6 x$ .

Frequently Asked Questions

How do you add polynomials?

Remove the parentheses and combine like terms by adding their coefficients. For example,

(3 x^{2} + 2 x) + (x^{2} - 5 x) = (3 + 1) x^{2} + (2 - 5) x = 4 x^{2} - 3 x

. Only terms with the same variable and exponent can be combined.

How do you subtract polynomials?

Distribute the minus sign to every term in the polynomial being subtracted, then combine like terms. For example,

(3 x^{2} + 2 x) - (x^{2} - 5 x) = 3 x^{2} + 2 x - x^{2} + 5 x = 2 x^{2} + 7 x

. The most common mistake is forgetting to change the sign of every term, not just the first one.

What does it mean to combine like terms?

Like terms have the same variable raised to the same exponent. To combine them, add or subtract their numerical coefficients and keep the variable part unchanged. For example,

4 x^{3} + (- 7 x^{3}) = (4 - 7) x^{3} = - 3 x^{3}

. Terms such as

5 x^{2}

and

5 x

are not like terms and cannot be combined.

When should I use the vertical method instead of the horizontal method?

Both methods give the same result. For short polynomials with few terms, the horizontal method is usually quicker. For longer polynomials with many terms, the vertical method is safer because aligning like terms in columns makes it harder to accidentally skip a term or drop a minus sign.

Does the order of addition matter for polynomials?

No. Polynomial addition is commutative:

P + Q = Q + P

. The result is the same regardless of order. Subtraction, however, is not commutative:

P - Q \neq Q - P

in general, so the order does matter for subtraction.

What happens when no like terms exist?

If two polynomials share no like terms, they cannot be simplified further. The result is simply all the terms written together. For example,

(x^{2} + 1) + (x^{3} + x) = x^{3} + x^{2} + x + 1

, since no terms of the same degree appear in both polynomials.

Quick Reference

The Rule

Adding Polynomials

Subtracting Polynomials

Vertical Method

Further Examples

Common Mistakes

Frequently Asked Questions

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