Equivalent Expressions Pattern - Equating Coefficients

Equating Coefficients

This pattern asks you to find an unknown constant by matching the structure of two equivalent expressions. The idea: if two polynomials are equal for all values of the variable, their corresponding coefficients must match. You might expand a product, simplify a fraction, or factor an expression, then compare it to the target form.

The Core Method

If $ax^2 + bx + c = dx^2 + ex + f$ for all $x$, then $a = d$, $b = e$, and $c = f$. Set corresponding coefficients equal and solve.

Worked Examples

Example 1. $(4y + 15)(7y - 11) = ky^2 + my + n$. What is the value of $m$?

Expand using FOIL:
First: $4y \cdot 7y = 28y^2$
Outer: $4y \cdot (-11) = -44y$
Inner: $15 \cdot 7y = 105y$
Last: $15 \cdot (-11) = -165$
Combine: $28y^2 + (-44y + 105y) - 165 = 28y^2 + 61y - 165$
Matching coefficients: $k = 28$, $m = 61$, $n = -165$.
The answer is $m = 61$.
Gotcha: The middle term comes from adding the Outer and Inner products. Don't drop the negative: $4y \cdot (-11) = -44y$, not $+44y$.

Example 2. $(7y^2 + 4) - (2y^2 - 9) = ky^2 + 13$. What is the value of $k$?

Simplify the left side:
$7y^2 + 4 - 2y^2 + 9 = 5y^2 + 13$
Match to $ky^2 + 13$: the $y^2$ coefficient is $k = 5$, and the constant $13 = 13$ checks out.
The answer is $k = 5$.
Gotcha: When subtracting the second polynomial, distribute the negative to every term inside: $-(2y^2 - 9) = -2y^2 + 9$. Forgetting to flip the $-9$ to $+9$ would give $k = 5$ but a wrong constant.

Example 3. $72w^5 + 16w^2 = aw^2(9w^3 + 2)$. What is the value of $a$?

The right side expands to $9aw^5 + 2aw^2$. Match coefficients with $72w^5 + 16w^2$:
$w^5$ term: $9a = 72$, so $a = 8$
$w^2$ term: $2a = 16$, so $a = 8$ ✓
The answer is $a = 8$.

Example 4. $\dfrac{155}{5x + 40}$ can be written as $\dfrac{a}{x + b}$ where $a$ and $b$ are constants. What is $b$?

Factor the denominator: $5x + 40 = 5(x + 8)$.
Simplify: $\dfrac{155}{5(x+8)} = \dfrac{31}{x + 8}$
Matching $\dfrac{a}{x + b}$: $a = 31$ and $b = 8$.
The answer is $b = 8$.
Gotcha: You must factor the denominator first. Don't try to "split" the fraction or cancel $5$ from $155$ and $5x$ separately — that doesn't work with addition in the denominator.

Example 5. If $6x^2 + cx - 35 = (mx + n)(x + p)$, where $c$, $m$, $n$, and $p$ are integer constants, which of the following must be an integer? A) $\dfrac{35}{n}$ B) $\dfrac{c}{m}$ C) $\dfrac{35}{m}$ D) $\dfrac{c}{n}$

Expand the right side: $(mx + n)(x + p) = mx^2 + mpx + nx + np = mx^2 + (mp + n)x + np$.
Match coefficients:
$x^2$: $m = 6$... but wait — $m$ doesn't have to be $6$. The factoring could be $(2x + n)(3x + p)$ or $(6x + n)(x + p)$ etc.
Constant term: $np = -35$
Since $n$ and $p$ are integers and $np = -35$, the value $\dfrac{35}{n}$ might not be an integer (for example, if $n = 7$ and $p = -5$, then $\dfrac{35}{7} = 5$ ✓, but the form doesn't guarantee $35/m$ is an integer).
The key: $np = -35$ means $p = -35/n$, so $35/n$ must be an integer. Actually more precisely, $|np| = 35$ and since $n$ is an integer factor of $35$ (up to sign), $35/n$ is always an integer.
The answer is A.
Gotcha: The constant-term equation $np = -35$ guarantees that $n$ divides $35$, making $35/n$ an integer. The other ratios depend on the specific factoring and are not guaranteed.

Example 6. The function $R(t) = \dfrac{36}{4t + 44}$ can be rewritten as $R(t) = \dfrac{a}{t + b}$. What are $a$ and $b$?

Factor the denominator: $4t + 44 = 4(t + 11)$.
Simplify: $\dfrac{36}{4(t+11)} = \dfrac{9}{t + 11}$
So $a = 9$ and $b = 11$.

Example 7. $(3x + c)(x + d) = 3x^2 + 17x + 20$. What is $cd$?

Expand: $3x^2 + 3dx + cx + cd = 3x^2 + (3d + c)x + cd$.
Match:
Middle term: $3d + c = 17$
Constant: $cd = 20$
From $cd = 20$: try factor pairs of $20$. If $c = 4, d = 5$: check $3(5) + 4 = 19$ ✗. If $c = 5, d = 4$: check $3(4) + 5 = 17$ ✓.
So $c = 5$, $d = 4$, and $cd = 20$.
The answer is $cd = 20$.

What to Do on Test Day

• Expand, then match. Multiply out the factored form, group by powers of the variable, and set each coefficient equal to the corresponding coefficient in the target expression.
• Factor denominators first. When a fraction is given in one form and asked for in another, factor the denominator to reveal the simplified structure.
• Use both equations as a cross-check. If matching gives you two equations that both determine the same unknown, verify your answer satisfies both.
• Integer divisibility questions: When the problem says "which must be an integer," focus on the constant-term equation. If $np = k$ and $n$ is an integer, then $k/n$ is automatically an integer.
• Middle-term sign errors: The FOIL middle term is the sum of Outer + Inner. Keep careful track of signs — a single sign flip changes the answer.
• Don't stop early. If the question asks for $a$ and you found $\dfrac{a}{8} = 1$, you still need to multiply: $a = 8$. Partial answers are common traps.