Consider the polynomial expression $P(x, y) = 5x^2y - 3xy^2 + 8x^2y - 11xy^2 + 4xy - 7x^2y$. After combining all like terms into the form $A x^2y + B xy^2 + C xy$, determine the numerical value of the ratio $\frac{A + B}{C}$.

Consider the polynomial expression $P(x, y) = 5x^2y - 3xy^2 + 8x^2y - 11xy^2 + 4xy - 7x^2y$ . After combining all like terms into the form $A x^2y + B xy^2 + C xy$ , determine the numerical value of the ratio $\frac{A + B}{C}$ .

Final answer

$-\frac{4}{1}$ (or $-2$ if evaluated as $\frac{6 - 14}{4} = -2$ )

Steps

Identify and group like terms

To simplify the polynomial and find the ratio $\frac{A + B}{C}$ , group and combine the like terms according to their variable parts and exponents.

Two terms are like terms if they have the exact same variables raised to the exact same powers:

Terms with $x^2y$ : $5x^2y$ , $+8x^2y$ , and $-7x^2y$
Terms with $xy^2$ : $-3xy^2$ and $-11xy^2$
Terms with $xy$ : $+4xy$

Rearrange the expression by grouping these sets of like terms together: $P(x, y) = (5x^2y + 8x^2y - 7x^2y) + (-3xy^2 - 11xy^2) + 4xy$

Common Pitfall: Be careful not to combine $x^2y$ and $xy^2$ . Although both contain the variables $x$ and $y$ , the exponents on each variable are different ( $x^2y$ has $x$ squared, whereas $xy^2$ has $y$ squared).

Combine the coefficients for each group

Apply the distributive property by adding or subtracting the numerical coefficients of each group:

For $x^2y$ : $(5 + 8 - 7)x^2y = 6x^2y \implies A = 6$
For $xy^2$ : $(-3 - 11)xy^2 = -14xy^2 \implies B = -14$
For $xy$ : $4xy \implies C = 4$

Thus, the simplified form is: $P(x, y) = 6x^2y - 14xy^2 + 4xy$

Compute the required ratio

Substitute the values $A = 6$ , $B = -14$ , and $C = 4$ into $\frac{A + B}{C}$ :

$\frac{A + B}{C} = \frac{6 + (-14)}{4} = \frac{-8}{4} = -2$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student ignores the negative signs when identifying coefficients, treating -3xy² and -11xy² as positive and obtaining B = 14.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
A = 5 + 8 - 7 = 6
Step 2
B = 3 + 11 = 14
Correct
B = -3 - 11 = -14
The terms involving $xy^2$ in the expression are $-3xy^2$ and $-11xy^2$ . The negative signs belong to the coefficients, so their sum is $(-3) + (-11) = -14$ , not $14$ .
Step 3
C = 4
Step 4
$\frac{(A + B)}{C}$ = $\frac{(6 + 14)}{4}$ = $\frac{20}{4}$ = 5
Correct
$\frac{A + B}{C}$ = $\frac{6 + (-14)}{4}$ = $\frac{-8}{4}$ = -2
Your arithmetic in this step followed logically from your value of $B = 14$ , but the result was altered because $B$ should have been $-14$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly distinguished the different variable groups ( $x^2y$ , $xy^2$ , and $xy$ ) without mixing unlike terms.

You accurately computed the combined coefficient for $x^2y$ as $A = 6$ .

You properly substituted your values into the target algebraic fraction and simplified the arithmetic.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign Error in Coefficient Combination

Dropping the leading minus signs on $-3xy^2$ and $-11xy^2$ when recording the coefficients to add.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating subtraction signs as mere separators between terms rather than as attached negative signs of the coefficients.

REASONING PATH

Your ACEpath teacher’s suggested approach

Attach the Sign to the Term: Every term in a polynomial carries the sign directly preceding it: $-axy^2$ means the coefficient is $-a$ .
Before adding coefficients, circle each term together with the plus or minus sign immediately in front of it.
Rewrite Subtraction as Addition of Negatives: Converting subtractions into additions makes the signs of the coefficients unmistakable: $P - Q = P + (-Q)$ .
Rewrite expressions like $-3xy^2 - 11xy^2$ as $+ (-3)xy^2 + (-11)xy^2$ before combining.

CORRECTED PATH

ACEpath Teacher's Demonstration

Identify the terms with $xy^2$ : $-3xy^2$ and $-11xy^2$ .
Combine their signed coefficients: $B = -3 + (-11) = -14$ .
Substitute $A = 6$ , $B = -14$ , and $C = 4$ into the ratio: $\frac{A + B}{C} = \frac{6 + (-14)}{4} = \frac{-8}{4} = -2$ .

Final answer

-2

NEXT STEPS

Final Takeaways from Your Teacher

Practice circling the sign in front of each term whenever you group like terms.
Solve the three targeted practice problems below to master combining terms with negative coefficients.

OverviewACEpath Teacher's Diagnosis

02 The student subtracts instead of adding negative coefficients when combining like terms for xy², computing -3 - (-11) = 8 instead of -3 - 11 = -14.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
A = 5 + 8 - 7 = 6
Step 2
B = -3 - (-11) = 8
Correct
B = -3 + (-11) = -3 - 11 = -14
The terms containing $xy^2$ in the polynomial are $-3xy^2$ and $-11xy^2$ . Their coefficients should be added together as $(-3) + (-11) = -3 - 11 = -14$ . By writing $-3 - (-11)$ , you introduced an extra negative sign and computed $-3 + 11 = 8$ .
Step 3
C = 4
Step 4
$\frac{(A + B)}{C}$ = $\frac{6 + 8}{4}$ = $\frac{14}{4}$ = $\frac{7}{2}$
Correct
$\frac{A + B}{C}$ = $\frac{6 + (-14)}{4}$ = $\frac{-8}{4}$ = -2
Your evaluation of the fraction was algebraically consistent with your value of $B = 8$ , but it led to an incorrect final result due to the sign error in Step 2.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly distinguished between $x^2y$ , $xy^2$ , and $xy$ as distinct sets of like terms.

You accurately combined the coefficients for the $x^2y$ terms to get $A = 6$ .

You correctly substituted your values into the target expression $\frac{A + B}{C}$ and simplified the resulting fraction.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign error when combining negative terms

You treated the subtraction sign in $-11xy^2$ as an operation subtracting a negative number $-(-11)$ , rather than taking the term's coefficient as $-11$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the sign of a term with an explicit subtraction operator: when adding terms $p + q$ where $p = -3$ and $q = -11$ , write $(-3) + (-11) = -3 - 11$ , not $-3 - (-11)$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Attach Signs to Coefficients: Every term carries its leading sign as part of its numerical coefficient: $-a x - b x = (-a)x + (-b)x = (-a - b)x$ .
Always read the terms with their preceding signs, sum them directly: $-3 - 11 = -14$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Combine the $xy^2$ terms: $-3xy^2 - 11xy^2 = (-3 - 11)xy^2 = -14xy^2$ , so $B = -14$ .
Substitute $A = 6$ , $B = -14$ , and $C = 4$ into $\frac{A + B}{C}$ .
Calculate the numerator: $A + B = 6 + (-14) = -8$ .
Divide by $C$ : $\frac{-8}{4} = -2$ .

Final answer

-2

NEXT STEPS

Final Takeaways from Your Teacher

Practice combining expressions with consecutive negative terms by treating each sign as belonging to the coefficient.
Verify integer additions with negative numbers using a number line or two-color counter model before substituting into composite formulas.

OverviewACEpath Teacher's Diagnosis

03 The student confuses x²y and xy² as identical like terms, combining all of them into a single term and incorrectly determining coefficients.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
(5 - 3 + 8 - 11 - 7)x^2y + 4xy = -8x^2y + 4xy
Correct
(5 + 8 - 7)x^2y + (-3 - 11)xy^2 + 4xy = 6x^2y - 14xy^2 + 4xy
Terms can only be combined if they have the exact same variables raised to the exact same powers. In $x^2y$ , the variable $x$ is squared, whereas in $xy^2$ , the variable $y$ is squared. These are distinct variable parts and must be grouped separately.
Step 2
$A = -8, \quad B = 0, \quad C = 4$
Correct
$A = 6, \quad B = -14, \quad C = 4$
Because $x^2y$ and $xy^2$ were collapsed into a single term, the coefficient of $xy^2$ was set to $0$ and $A$ received the sum of all five coefficients.
Step 3
$\frac{A + B}{C}$ = $\frac{-8 + 0}{4}$ = -2
Correct
$\frac{A + B}{C}$ = $\frac{6 + (-14)}{4}$ = $\frac{-8}{4}$ = -2
You substituted your values into the ratio and calculated the arithmetic correctly, which coincidentally matched the correct numerical value of $-2$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly recognized that $4xy$ is separate from the higher-degree terms.

You performed the addition and subtraction of the signed integer coefficients accurately.

You substituted your extracted coefficients into the required ratio formula without arithmetic errors.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Confusing like terms with swapped exponents

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated terms with the same variable letters as like terms without matching which exponent belongs to which variable ( $x^2y$ vs. $xy^2$ ).

REASONING PATH

Your ACEpath teacher’s suggested approach

Variable and Exponent Match Rule: Two terms are like terms if and only if each variable appears with the exact same exponent in both terms.
Before adding coefficients, compare the exponent on every individual variable: $x^2y = x \cdot x \cdot y$ , while $xy^2 = x \cdot y \cdot y$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Group $x^2y$ terms: $(5 + 8 - 7)x^2y = 6x^2y$ , so $A = 6$ .
Group $xy^2$ terms: $(-3 - 11)xy^2 = -14xy^2$ , so $B = -14$ .
Identify the $xy$ term: $4xy$ , so $C = 4$ .
Compute the ratio: $\frac{A + B}{C} = \frac{6 + (-14)}{4} = \frac{-8}{4} = -2$ .

Final answer

-2

NEXT STEPS

Final Takeaways from Your Teacher

Underline or color-code like terms by matching both the variable and its specific power before collecting coefficients.
Practice distinguishing pairs like $a^2b$ and $ab^2$ , or $u^3v$ and $uv^3$ , to make exponent verification automatic.

Lock it in

Question

In the polynomial $7x - 4y - 9x$ , what are the signed coefficients of the like terms containing $x$ ?

Answer

$7$ and $-9$ . The minus sign preceding $9x$ belongs to its coefficient: $7x + (-9)x = -2x$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student ignores the negative signs when identifying coefficients, treating -3xy² and -11xy² as positive and obtaining B = 14.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student subtracts instead of adding negative coefficients when combining like terms for xy², computing -3 - (-11) = 8 instead of -3 - 11 = -14.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student confuses x²y and xy² as identical like terms, combining all of them into a single term and incorrectly determining coefficients.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

Similar question