Adding and Subtracting Polynomials

What a polynomial is

A polynomial is a sum of terms. Each term is a number multiplied by a whole-number power of the variable.

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

Part	Name
$3x^2$, $-5x$, $2$	the terms
$3$, $-5$, $2$	the coefficients
$2$	the degree — the highest power present
$3$	the leading coefficient — the one on the highest power

The sign in front belongs to the term. In $3x^2 - 5x$ the second term is $-5x$, not $5x$, and carrying that sign along is what keeps the later steps right.

A few expressions look close but are not polynomials:

Expression	Why not
$\tfrac{3}{x}$	dividing by the variable is a power of $-1$
$\sqrt{x}$	that is a power of $\tfrac{1}{2}$, not a whole number
$2^x$	the variable is in the exponent, not the base

Only like terms combine

Like terms have the same variable raised to the same power.

$$4x^2 + 7x^2 = 11x^2$$ $$4x^2 + 7x \;\text{ stays as it is}$$

The reason is the one from combining like terms: a term counts something, and $x^2$ and $x$ count different things. Four squares plus seven lengths is not eleven of anything.

Substituting a number makes it concrete. At $x = 3$:

$$4(9) + 7(3) = 36 + 21 = 57$$

while $11x^2$ would give $99$ and $11x$ would give $33$. Neither matches, which is what “these do not combine” means.

Adding

Collect the like terms and add their coefficients.

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

Lining the powers up in columns keeps them together:

Power	First	Second	Sum
$x^2$	$3$	$1$	$4$
$x$	$-5$	$4$	$-1$
constant	$2$	$-7$	$-5$

$$= 4x^2 - x - 5$$

Note the missing coefficient trap: $x^2$ on its own has coefficient $1$, not $0$.

Why subtracting needs the minus distributed

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

The minus sign is in front of the whole set of parentheses, so it applies to everything inside them:

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

Every sign flipped, including the $-7$, which became $+7$.

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

Changing only the first sign is the most common error in all of algebra. It is worth writing the flipped parentheses on their own line before adding, so the sign change happens once and is visible.

Check it with a number. At $x = 1$ the first polynomial is $0$ and the second is $-2$, so the difference should be $2$:

$$2(1) - 9(1) + 9 = 2 \quad\checkmark$$

A single substitution catches a dropped sign in seconds.

The answer is always a polynomial

Add two polynomials and you get a polynomial. Subtract and you get a polynomial. Multiply, and the same holds.

Nothing in these operations can produce a negative power or a root, because adding coefficients leaves the powers untouched. Polynomials are closed under addition, subtraction and multiplication.

This is the same property the integers have: adding or multiplying whole numbers never takes you outside the whole numbers, while dividing can. Division is the one that breaks it for polynomials too — $\tfrac{x}{x^2}$ is not a polynomial.

Worked examples

Common mistakes

Practice problems