Adding, Subtracting, and Multiplying Polynomials

Recommended resources

The freely available resources this lesson will direct you to.

Polynomials - Adding, Subtracting, Multiplying and Dividing Algebraic Expressionswww.youtube.com
More examples of addition and subtraction of polynomials | Algebra II | Khan Academywww.youtube.com
Operations with polynomials | Lesson (article) | Khan Academywww.khanacademy.org
5.1 Add and Subtract Polynomials - Intermediate Algebra 2eopenstax.org

Welcome back. In the previous lesson, you practiced exponent rules, including the key fact that when multiplying powers with the same base, you add exponents. That rule now becomes part of a larger process: multiplying entire polynomial expressions.

In this lesson, you will add, subtract, and multiply polynomials accurately. The main habits are to identify like terms, handle subtraction signs carefully, distribute every factor during multiplication, and then write the result in standard form. These skills are essential preparation for factoring and solving quadratic equations in the next lessons.

Polynomial structure and like terms

A polynomial is an expression made by adding or subtracting terms whose variable exponents are nonnegative whole numbers. For example,

4 x^{3} - 7 x + 2

is a polynomial with three terms:

4 x^{3}, - 7 x, 2

The number multiplying the variable part is the coefficient. Thus, $4$ is the coefficient of $x^{3}$ , and $- 7$ is the coefficient of $x$ . A number with no variable, such as $2$ , is a constant.

Polynomials receive names according to their number of terms:

Number of terms	Name	Example
1	monomial	$5 x^{2}$
2	binomial	$x - 6$
3	trinomial	$2 x^{2} + x - 4$

The central idea for addition and subtraction is like terms. Terms are like terms only when their variable parts match exactly, including every exponent.

3 x^{2} and - 8 x^{2}

are like terms, because both contain $x^{2}$ . Their coefficients can be combined:

3 x^{2} - 8 x^{2} = - 5 x^{2}

But these are not like terms:

x^{2} and x^{3}

ab and a^{2} b

a b^{2} and a^{2} b

The coefficients may be different, but the variable-and-exponent pattern must be identical. You can think of $x^{2}$ , $x$ , and constants as different categories: coefficients can only be added or subtracted within the same category.

Operations with polynomials | Lesson (article) | Khan Academy

Read Khan Academy’s explanation of adding and subtracting polynomials. It reinforces the exact meaning of like terms and highlights the sign change required when subtracting an entire polynomial.

In “How do I add and subtract polynomials?” and “What should I be careful of when adding and subtracting polynomials?”, read the like-term explanation. Then continue through the numbered procedure and the two worked binomial examples. Focus especially on the example beginning with 4x^2-7x+1: notice that subtracting parentheses means every term inside changes sign.

Adding and subtracting polynomials

Addition: remove parentheses, then combine

When a polynomial is added, the parentheses mainly group the expression. You may remove them without changing any signs.

Consider:

(5 x^{2} - 3 x + 8) + (2 x^{2} + 7 x - 5)

First rewrite without parentheses:

5 x^{2} - 3 x + 8 + 2 x^{2} + 7 x - 5

Group terms of the same type:

(5 x^{2} + 2 x^{2}) + (- 3 x + 7 x) + (8 - 5)

Then combine the coefficients:

7 x^{2} + 4 x + 3

A good final form lists terms in descending powers of $x$ :

7 x^{2} + 4 x + 3

Subtraction: distribute the negative sign first

Subtraction is more dangerous because the negative sign applies to the entire polynomial after it.

Consider:

(6 x^{2} - 4 x + 9) - (2 x^{2} + 5 x - 3)

Rewrite subtraction as adding the opposite:

6 x^{2} - 4 x + 9 - 2 x^{2} - 5 x + 3

Every sign in the second polynomial changed:

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

became

- 2 x^{2} - 5 x + 3

Now combine like terms:

(6 x^{2} - 2 x^{2}) + (- 4 x - 5 x) + (9 + 3)

= 4 x^{2} - 9 x + 12

Therefore,

(6 x^{2} - 4 x + 9) - (2 x^{2} + 5 x - 3) = 4 x^{2} - 9 x + 12

A reliable subtraction routine is:

Keep the first polynomial unchanged.
Change the subtraction sign to addition.
Change every sign inside the second parentheses.
Combine like terms.

Multiple variables: match all variable parts

With more than one variable, like terms still must match perfectly.

Simplify:

3 a^{2} b - 4 a b^{2} + 6 a^{2} b + 2 a b^{2} - 5 b

Group matching terms:

(3 a^{2} b + 6 a^{2} b) + (- 4 a b^{2} + 2 a b^{2}) - 5 b

Then combine:

9 a^{2} b - 2 a b^{2} - 5 b

Notice that $a^{2} b$ and $a b^{2}$ cannot combine. Both use $a$ and $b$ , but their exponents are assigned differently.

Multiplying polynomial terms

Multiplication works differently from addition. When multiplying terms, multiply coefficients and use the exponent rule for any matching variable bases.

For example:

(3 x^{2}) (- 4 x^{5})

Multiply the coefficients:

3 (- 4) = - 12

Then add the exponents on the shared base $x$ :

x^{2} \cdot x^{5} = x^{2 + 5} = x^{7}

So:

(3 x^{2}) (- 4 x^{5}) = - 12 x^{7}

With multiple variables, apply the rule separately to each matching base:

(- 2 a^{3} b) (5 a^{2} b^{4})

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

- 10 a^{5} b^{5}

Be careful not to confuse multiplying terms with adding terms:

x^{2} + x^{3}

cannot be combined because these are unlike terms. However,

x^{2} \cdot x^{3} = x^{5}

because this is multiplication of matching bases.

Distributing to multiply polynomials

To multiply a monomial by a polynomial, distribute the monomial to every term inside the parentheses.

Consider:

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

Multiply $- 3 x$ by each term:

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

Simplify each product:

- 6 x^{3} + 3 x^{2} - 12 x

Thus,

- 3 x (2 x^{2} - x + 4) = - 6 x^{3} + 3 x^{2} - 12 x

Nothing needs to combine here because the three resulting terms have different powers of $x$ .

For two polynomials, every term in one polynomial must multiply every term in the other. A useful accuracy check is to count expected initial products:

Factors	Initial products
monomial and trinomial	$1 \cdot 3 = 3$
binomial and binomial	$2 \cdot 2 = 4$
binomial and trinomial	$2 \cdot 3 = 6$
trinomial and trinomial	$3 \cdot 3 = 9$

If you multiply a binomial and a trinomial but write only five products before combining terms, one multiplication was probably missed.

Polynomials - Adding, Subtracting, Multiplying and Dividing Algebraic Expressions

Watch “Polynomials - Adding, Subtracting, Multiplying and Dividing Algebraic Expressions” by The Organic Chemistry Tutor for a visual walkthrough of the core procedures used in this lesson.

Watch addition and subtraction. Focus on how like terms are grouped and how a subtraction sign changes each term in the polynomial that follows it. Then watch polynomial multiplication, which moves from FOIL for binomials to distributing a binomial across a trinomial. Pause briefly before each example is completed and check whether you accounted for every product.

Multiplying two binomials: FOIL is organized distribution

A binomial has two terms, so multiplying two binomials produces four initial products. The mnemonic FOIL helps organize these four products:

First terms
Outer terms
Inner terms
Last terms

FOIL is not a separate rule from distribution. It is simply a convenient checklist for the special case of two binomials.

The image applies FOIL to $(y-7)(y+4)$: it multiplies first, outer, inner, and last term pairs, then combines $4y$ and $-7y$ to obtain $y^2-3y-28$.

Now multiply:

(2 x - 3) (x + 5)

Write all four products:

(2 x) (x) + (2 x) (5) + (- 3) (x) + (- 3) (5)

Simplify each product:

2 x^{2} + 10 x - 3 x - 15

The middle terms are like terms:

10 x - 3 x = 7 x

So the product is:

2 x^{2} + 7 x - 15

The same method works when signs are negative:

(x - 4) (x - 6)

= x (x) + x (- 6) + (- 4) (x) + (- 4) (- 6)

= x^{2} - 6 x - 4 x + 24

x^{2} - 10 x + 24

The final positive constant is not accidental:

(- 4) (- 6) = 24

Squaring a binomial means multiplying it by itself

A common error is to square each term separately and forget the middle products. For instance,

(x + 3)^{2}

means

(x + 3) (x + 3)

Apply distribution:

x^{2} + 3 x + 3 x + 9

(x + 3)^{2} = x^{2} + 6 x + 9

It is not correct to write:

(x + 3)^{2} = x^{2} + 9

That incorrect expression omits the two cross-products, $3 x$ and $3 x$ .

Operations with polynomials | Lesson (article) | Khan Academy

Return to Khan Academy for its concise account of polynomial multiplication. This section connects distribution, FOIL, and exponent rules in one procedure.

Under “How do I multiply polynomials?” read the multiplication procedure. Pay particular attention to the statement that every term in one polynomial must be distributed to all terms in the other. Then follow the examples involving (2x-1)(x-5) and 3x(x^2-4x+9), checking where exponent addition occurs and where like terms are combined afterward.

Beyond FOIL: a binomial times a trinomial

FOIL is useful only for two binomials. For any other pair of polynomials, use ordinary distribution systematically.

Multiply:

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

First distribute $3 x$ across all three terms:

3 x (x^{2} + 4 x - 5) = 3 x^{3} + 12 x^{2} - 15 x

Then distribute $- 2$ across all three terms:

- 2 (x^{2} + 4 x - 5) = - 2 x^{2} - 8 x + 10

Put the two results together:

3 x^{3} + 12 x^{2} - 15 x - 2 x^{2} - 8 x + 10

Finally, combine like terms:

3 x^{3} + 10 x^{2} - 23 x + 10

The structure matters more than speed:

Make every required product.
Simplify each product using integer and exponent rules.
Combine like terms only after all products are visible.
Write the final polynomial in descending order of degree.

A quick way to verify the multiplication is to substitute a simple value, such as $x = 1$ .

Original expression:

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

= (1) (0) = 0

Simplified result:

3 (1)^{3} + 10 (1)^{2} - 23 (1) + 10

= 3 + 10 - 23 + 10 = 0

Both expressions give the same value, so this check supports the result. It does not replace showing the algebra, but it can catch sign errors.

Common errors and how to prevent them

Error	Why it fails	Reliable correction
Combining $x^{2}$ and $x$	Their exponents differ.	Combine only terms with identical variable parts.
Leaving signs unchanged after subtraction	The negative sign applies to the whole second polynomial.	Rewrite subtraction as addition of opposites.
Multiplying only first terms	Distribution requires every term pair.	Count the expected products first.
Writing $(x + 4)^{2} = x^{2} + 16$	Two cross-products were omitted.	Rewrite it as $(x + 4) (x + 4)$ and distribute.
Adding exponents while adding terms	Exponent addition belongs to multiplication, not addition.	Use exponent rules only within a product of matching bases.
Combining terms before distribution is complete	Missing products can change the entire result.	Expand first; combine like terms last.

Key takeaways

A polynomial is built from terms with nonnegative whole-number exponents. To work with polynomials accurately:

Add and subtract by combining only like terms.
During subtraction, distribute the negative sign to every term in the polynomial being subtracted.
Multiply terms by multiplying coefficients and adding exponents of matching bases.
Multiply polynomials by distributing every term in one factor to every term in the other.
FOIL is a checklist for multiplying two binomials, not a different operation.
After multiplying, combine like terms and write the result in descending powers.

Next, you will reverse this process by factoring polynomial expressions, beginning with greatest common factors and common quadratic patterns.