Expand and Combine

Expanding and Combining Like Terms

This pattern is about simplifying polynomial expressions. You might need to combine like terms, distribute a coefficient across parentheses, or multiply two binomials using FOIL. The key rule: you can only add or subtract terms that have the exact same variable part.

The Core Rules

Like terms share the same variable(s) raised to the same power(s). To combine them, add or subtract the coefficients and keep the variable part unchanged.

$7y^2 + 8y^2 = 15y^2$ — add the coefficients, the exponent stays $2$
$15p^7 - 6p^7 = 9p^7$ — subtract the coefficients, the exponent stays $7$

The distributive property lets you remove parentheses by multiplying each term inside by the factor outside:

$9(y + 12) = 9y + 108$

When multiplying two binomials, use FOIL (First, Outer, Inner, Last) and then combine the middle terms:

$\left(\dfrac{1}{2}x + 5\right)(4x - 6)$: First $= 2x^2$, Outer $= -3x$, Inner $= 20x$, Last $= -30$, so the result is $2x^2 + 17x - 30$

Worked Examples

Example 1. Which expression is equivalent to $7y^2 + 8y^2$?

A) $y^2$
B) $15y^4$
C) $15y^2$
D) $y^4$

Both terms have the variable part $y^2$, so they are like terms. Add the coefficients: $7 + 8 = 15$. The variable part does not change.
$7y^2 + 8y^2 = 15y^2$
Gotcha: Option B ($15y^4$) adds the exponents. When combining like terms, you never change the exponent — you only combine coefficients.
The answer is C.

Example 2. Which expression is equivalent to $18x - (5x + 2x)$?

A) $15x$
B) $8x$
C) $11x$
D) $25x$

First simplify inside the parentheses: $5x + 2x = 7x$.
Then subtract: $18x - 7x = 11x$.
Gotcha: If you forget that the subtraction applies to the entire group, you might compute $18x - 5x + 2x = 15x$ (option A). The parentheses mean you subtract the whole sum.
The answer is C.

Example 3. Which expression is equivalent to $(4y^3 + 8y^2 - 3) + (11y^3 + 6y + 1)$?

A) $15y^6 + 8y^2 + 6y - 2$
B) $15y^3 + 8y^2 + 6y - 2$
C) $15y^3 + 14y^2 - 2$
D) $15y^3 + 8y^2 + 6y - 3$

Group by matching variable parts:
$y^3$ terms: $4y^3 + 11y^3 = 15y^3$
$y^2$ term: $8y^2$ (no match in the second polynomial)
$y$ term: $6y$ (no match in the first polynomial)
Constants: $-3 + 1 = -2$
Result: $15y^3 + 8y^2 + 6y - 2$
Gotcha: Option A adds exponents ($y^3 + y^3 = y^6$). Option C combines unlike terms ($8y^2 + 6y = 14y^2$). You can only combine terms with the exact same power.
The answer is B.

Example 4. $p(x) = \dfrac{1}{2}x + 5$ and $q(x) = 4x - 6$. Which expression is equivalent to $p(x) \cdot q(x)$?

A) $2x^2 - 30$
B) $2x^2 + 17x - 30$
C) $2x^2 - 23x - 30$
D) $4x^2 + 17x - 30$

Use FOIL on $\left(\dfrac{1}{2}x + 5\right)(4x - 6)$:
First: $\dfrac{1}{2}x \cdot 4x = 2x^2$
Outer: $\dfrac{1}{2}x \cdot (-6) = -3x$
Inner: $5 \cdot 4x = 20x$
Last: $5 \cdot (-6) = -30$
Combine the middle terms: $-3x + 20x = 17x$
Result: $2x^2 + 17x - 30$
Gotcha: Option A only multiplies First and Last, skipping the middle terms entirely. Option D gets $\dfrac{1}{2} \times 4 = 4$ instead of $2$. Always multiply the coefficients carefully when fractions are involved.
The answer is B.

Example 5. $f(a) = \dfrac{2}{3}a - 4$ and $g(a) = 9a + 3$. Which expression is equivalent to $f(a) \cdot g(a)$?

A) $6a^2 + 38a - 12$
B) $6a^2 - 12$
C) $\dfrac{2}{27}a^2 - 34a - 12$
D) $6a^2 - 34a - 12$

FOIL on $\left(\dfrac{2}{3}a - 4\right)(9a + 3)$:
First: $\dfrac{2}{3}a \cdot 9a = \dfrac{18}{3}a^2 = 6a^2$
Outer: $\dfrac{2}{3}a \cdot 3 = 2a$
Inner: $(-4)(9a) = -36a$
Last: $(-4)(3) = -12$
Middle terms: $2a + (-36a) = -34a$
Result: $6a^2 - 34a - 12$
Gotcha: Option A flips the sign on the middle term ($2a + 36a = 38a$). Watch the negative on the $-4$ — it makes the inner product negative.
The answer is D.

Example 6. Which expression is the simplified form of $6k^7 + 13k^7$?

A) $-7k^7$
B) $19k^{14}$
C) $-7k^{14}$
D) $19k^7$

Both terms have $k^7$. Add coefficients: $6 + 13 = 19$. Exponent stays $7$.
$6k^7 + 13k^7 = 19k^7$
The answer is D.

What to Do on Test Day

• Like terms rule: Only terms with the exact same variable and exponent can be combined. $3x^2$ and $5x$ are NOT like terms.
• Combining like terms: Add or subtract the coefficients. The variable part (including the exponent) never changes. $7y^2 + 8y^2 = 15y^2$, not $15y^4$.
• Distributive property: Multiply the outside factor by every term inside: $a(b + c) = ab + ac$.
• Subtracting a group: A minus sign in front of parentheses flips every sign inside: $-(5x + 2x) = -5x - 2x$.
• FOIL for binomials: Multiply First, Outer, Inner, Last, then combine the two middle terms. Don't skip the middle — that's the most common error.
• Fractions in FOIL: When a binomial has a fractional coefficient, multiply numerator × numerator carefully. $\dfrac{1}{2} \times 4 = 2$, not $4$.
• Quick check: If the answer choices differ by exponents (like $y^2$ vs. $y^4$), it's testing whether you add exponents (wrong) or keep them (right).