Table of Contents
Adding polynomials
In the parent chapter, polynomials and their terms (coefficients, variables, exponents, degree) are already introduced. Here we focus on how to combine polynomials using the basic algebraic operations, starting with addition.
To add polynomials, you combine like terms. Like terms have:
- The same variable part, including exponents (for example, $3x^2$ and $-5x^2$ are like terms; $3x^2$ and $3x^3$ are not).
The key idea: add coefficients of like terms; keep the variable part the same.
Example:
$$
(3x^2 + 4x - 7) + (5x^2 - 2x + 10)
$$
Group like terms:
$$
(3x^2 + 5x^2) + (4x - 2x) + (-7 + 10)
$$
Compute:
$$
8x^2 + 2x + 3
$$
You can do this “horizontally” as above, or by stacking polynomials so like terms are in the same column, similar to adding multi-digit numbers.
Example (stacked layout):
$$
\begin{aligned}
&(2x^3 + 5x^2 - 4x + 1) \\
&\phantom{+}(3x^3 - x^2 + 6x - 8) \\
\hline
&5x^3 + 4x^2 + 2x - 7
\end{aligned}
$$
Notice you never change the exponents when adding; you only add the numeric coefficients.
Subtracting polynomials
Subtracting polynomials is closely related to addition. You can think of subtraction as:
“Add the opposite.”
To subtract a polynomial, you distribute a negative sign across the second polynomial and then add.
Example:
$$
(4x^2 - 3x + 5) - (x^2 + 2x - 7)
$$
- Distribute the minus sign to every term in the second polynomial:
$$
(4x^2 - 3x + 5) + (-x^2 - 2x + 7)
$$ - Group like terms:
$$
(4x^2 - x^2) + (-3x - 2x) + (5 + 7)
$$ - Compute:
$$
3x^2 - 5x + 12
$$
You must be careful with signs: the minus sign outside parentheses changes every sign inside.
Common error example:
Incorrect:
$$
(2x^2 + 3x - 1) - (x^2 - 4x + 5)
= 2x^2 + 3x - 1 - x^2 - 4x + 5
$$
Here the last two terms were not both negated.
Correct:
Distribute the minus sign to each term:
$$
(2x^2 + 3x - 1) + (-x^2 + 4x - 5)
= (2x^2 - x^2) + (3x + 4x) + (-1 - 5)
= x^2 + 7x - 6
$$
Multiplying a monomial and a polynomial
A monomial is a single term like $5x$, $-3x^2$, or $\frac{1}{2}x^3y$.
To multiply a monomial by a polynomial, use the distributive property:
$$
a(b + c + d) = ab + ac + ad
$$
So, for a monomial $m$ and a polynomial $P(x)$,
$$
m \cdot P(x) = m \cdot \text{(each term of } P(x)\text{)}
$$
Example:
$$
3x(2x^2 - x + 4)
$$
Multiply $3x$ by each term:
- $3x \cdot 2x^2 = 6x^3$ (multiply coefficients $3 \cdot 2$, add exponents: $x^{1+2}$)
- $3x \cdot (-x) = -3x^2$
- $3x \cdot 4 = 12x$
So,
$$
3x(2x^2 - x + 4) = 6x^3 - 3x^2 + 12x
$$
Another example with negative and fractional coefficients:
$$
-\frac{1}{2}x^2(4x^3 - 6x + 1)
$$
Multiply term-by-term:
- $-\frac{1}{2}x^2 \cdot 4x^3 = -2x^5$
- $-\frac{1}{2}x^2 \cdot (-6x) = 3x^3$
- $-\frac{1}{2}x^2 \cdot 1 = -\frac{1}{2}x^2$
Result:
$$
-2x^5 + 3x^3 - \frac{1}{2}x^2
$$
Multiplying two polynomials
When you multiply two polynomials, each term in the first polynomial multiplies each term in the second, and then you combine like terms.
In general, if
$$
P(x) = a_0 + a_1x + \dots + a_mx^m,
\quad
Q(x) = b_0 + b_1x + \dots + b_nx^n,
$$
then every product $a_ix^i \cdot b_jx^j = (a_ib_j)x^{i+j}$ appears in the product, and like terms are combined.
Binomial times binomial (FOIL pattern)
When both polynomials have exactly two terms (binomials), a common pattern is often called FOIL:
- First terms
- Outer terms
- Inner terms
- Last terms
Example:
$$
(x + 3)(x + 5)
$$
Multiply:
- First: $x \cdot x = x^2$
- Outer: $x \cdot 5 = 5x$
- Inner: $3 \cdot x = 3x$
- Last: $3 \cdot 5 = 15$
Add:
$$
x^2 + 5x + 3x + 15 = x^2 + 8x + 15
$$
The FOIL idea is just a specific case of “every term times every term”; it does not apply directly when there are more than two terms in either factor, but the general idea still does.
Example with negatives:
$$
(2x - 7)(x + 4)
$$
- First: $2x \cdot x = 2x^2$
- Outer: $2x \cdot 4 = 8x$
- Inner: $-7 \cdot x = -7x$
- Last: $-7 \cdot 4 = -28$
Combine:
$$
2x^2 + 8x - 7x - 28 = 2x^2 + x - 28
$$
General binomial times polynomial
For a binomial times a polynomial with more than two terms, you can think of distributing in two stages.
Example:
$$
(x + 2)(3x^2 - x + 5)
$$
Distribute $(x + 2)$:
$$
(x + 2)(3x^2 - x + 5)
= x(3x^2 - x + 5) + 2(3x^2 - x + 5)
$$
Now multiply each:
- $x(3x^2 - x + 5) = 3x^3 - x^2 + 5x$
- $2(3x^2 - x + 5) = 6x^2 - 2x + 10$
Add:
$$
3x^3 - x^2 + 5x + 6x^2 - 2x + 10
= 3x^3 + 5x^2 + 3x + 10
$$
Polynomial times polynomial (more terms)
The same principle works for any polynomials.
Example:
$$
(2x^2 + 3x - 1)(x^2 - 4)
$$
You can treat this as multiplying by each term of the second polynomial:
- Multiply by $x^2$:
$$
x^2(2x^2 + 3x - 1) = 2x^4 + 3x^3 - x^2
$$ - Multiply by $-4$:
$$
-4(2x^2 + 3x - 1) = -8x^2 - 12x + 4
$$ - Add the results:
$$
2x^4 + 3x^3 - x^2 - 8x^2 - 12x + 4
= 2x^4 + 3x^3 - 9x^2 - 12x + 4
$$
Another example:
$$
(x^2 + 2x + 3)(x - 5)
$$
Distribute $x - 5$:
$$
x(x^2 + 2x + 3) - 5(x^2 + 2x + 3)
$$
Compute:
$$
= (x^3 + 2x^2 + 3x) + (-5x^2 - 10x - 15)
$$
Combine like terms:
$$
x^3 - 3x^2 - 7x - 15
$$
Special product patterns
When multiplying certain binomials, the result follows recognizable patterns. Knowing these can save time and help later with factoring.
Square of a binomial
For any expressions $a$ and $b$:
$$
(a + b)^2 = a^2 + 2ab + b^2
$$
$$
(a - b)^2 = a^2 - 2ab + b^2
$$
You can verify these by multiplying:
$$
(a + b)^2 = (a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2
$$
Examples:
- $(x + 4)^2 = x^2 + 8x + 16$
- $(3x - 5)^2 = 9x^2 - 30x + 25$
Product of sum and difference
For any $a$ and $b$:
$$
(a + b)(a - b) = a^2 - b^2
$$
This is called the difference of squares pattern.
Example:
$$
(x + 7)(x - 7) = x^2 - 49
$$
since $a = x$, $b = 7$.
Another example:
$$
(5x + 2)(5x - 2) = 25x^2 - 4
$$
These patterns are part of “special products” and are especially useful in recognizing and simplifying later expressions; they are also closely connected with factoring.
Dividing a polynomial by a monomial
Full polynomial division methods (such as long division and synthetic division) are treated elsewhere. Here we handle the simpler case: dividing a polynomial by a single-term monomial.
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial, as long as you are dividing by a nonzero expression.
Example:
$$
\frac{6x^3 - 9x^2 + 3x}{3x}
$$
Divide term by term:
- $\dfrac{6x^3}{3x} = 2x^2$
- $\dfrac{-9x^2}{3x} = -3x$
- $\dfrac{3x}{3x} = 1$
So:
$$
\frac{6x^3 - 9x^2 + 3x}{3x} = 2x^2 - 3x + 1
$$
Another example with exponents reducing:
$$
\frac{10x^4y^2 - 5x^2y}{5x^2y}
$$
Divide each term:
- $\dfrac{10x^4y^2}{5x^2y} = 2x^{4-2}y^{2-1} = 2x^2y$
- $\dfrac{-5x^2y}{5x^2y} = -1$
So:
$$
\frac{10x^4y^2 - 5x^2y}{5x^2y} = 2x^2y - 1
$$
Be careful not to divide only some terms; each term in the numerator must be divided by the monomial in the denominator.
Incorrect:
$$
\frac{6x^3 - 9x^2 + 3x}{3x} = 2x^2 - 9x^2 + 3x
$$
Here the division was not applied to each term. The correct method always separates the fraction across the sum:
$$
\frac{6x^3 - 9x^2 + 3x}{3x}
= \frac{6x^3}{3x} - \frac{9x^2}{3x} + \frac{3x}{3x}
$$
Simplifying polynomial expressions
Many problems combine multiple operations. The goal is to rewrite the expression in a standard, simplified polynomial form:
- No parentheses (unless required by context).
- Like terms combined.
- Terms ordered by decreasing exponent (often called standard form).
General steps:
- Remove parentheses using distribution (and pay attention to minus signs).
- Multiply as needed (monomial–polynomial or polynomial–polynomial).
- Combine like terms.
- Order terms by degree, usually highest to lowest.
Example:
Simplify
$$
(2x - 3)(x + 4) - (x^2 - 5x - 6).
$$
Step 1: Multiply the binomials:
- $(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$
Step 2: Subtract the second polynomial by distributing the minus sign:
$$
2x^2 + 5x - 12 - (x^2 - 5x - 6)
= 2x^2 + 5x - 12 - x^2 + 5x + 6
$$
Step 3: Combine like terms:
- $2x^2 - x^2 = x^2$
- $5x + 5x = 10x$
- $-12 + 6 = -6$
Final simplified form:
$$
x^2 + 10x - 6
$$
Another example:
Simplify
$$
3(x^2 - 2x + 1) + 2(x^2 + x - 4).
$$
Distribute:
$$
= (3x^2 - 6x + 3) + (2x^2 + 2x - 8)
$$
Combine like terms:
$$
(3x^2 + 2x^2) + (-6x + 2x) + (3 - 8)
= 5x^2 - 4x - 5
$$
Being careful with signs, exponents, and systematically combining like terms is at the core of working with polynomial operations.