A student simplified the expression $(ax^3 - 5x^2 + 7x) - (4x^3 + bx^2 - cx + 9) + (2x^3 + 3x^2 - 8)$ by combining like terms and obtained the final result $5x^3 - 6x^2 + 19x - 17$. Find the original constant values of $a$, $b$, and $c$.

A student simplified the expression $(ax^3 - 5x^2 + 7x) - (4x^3 + bx^2 - cx + 9) + (2x^3 + 3x^2 - 8)$ by combining like terms and obtained the final result $5x^3 - 6x^2 + 19x - 17$ . Find the original constant values of $a$ , $b$ , and $c$ .

Final answer

$a = 7$ , $b = 4$ , $c = 12$

Steps

Distribute signs and remove parentheses

To find the values of $a$ , $b$ , and $c$ , expand the expression by removing the parentheses and then group like terms according to their degree.

Pay close attention to the negative sign in front of the second set of parentheses, which flips the sign of every term inside: $-(4x^3 + bx^2 - cx + 9) = -4x^3 - bx^2 + cx - 9$

Rewriting the entire expression without parentheses: $ax^3 - 5x^2 + 7x - 4x^3 - bx^2 + cx - 9 + 2x^3 + 3x^2 - 8$

Common Pitfall: Forgetting to distribute the negative sign to the term $-cx$ is a frequent mistake. It becomes $-(-cx) = +cx$ , not $-cx$ .

Group like terms

Group the terms by the power of $x$ :

$x^3$ terms: $ax^3 - 4x^3 + 2x^3 = (a - 4 + 2)x^3 = (a - 2)x^3$
$x^2$ terms: $-5x^2 - bx^2 + 3x^2 = (-5 - b + 3)x^2 = (-2 - b)x^2$
$x$ terms: $7x + cx = (7 + c)x$
Constant terms: $-9 - 8 = -17$

Combining these yields the simplified expression: $(a - 2)x^3 + (-2 - b)x^2 + (7 + c)x - 17$

Match coefficients with the given result

The problem states that the simplified expression equals $5x^3 - 6x^2 + 19x - 17$ . Equate corresponding coefficients:

For the $x^3$ term: $a - 2 = 5 \implies a = 7$
For the $x^2$ term: $-2 - b = -6 \implies -b = -4 \implies b = 4$
For the $x$ term: $7 + c = 19 \implies c = 12$

The constant term matches: $-17 = -17$ .

Thus, the values are $a = 7$ , $b = 4$ , and $c = 12$ .

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the negative sign to the second and third terms inside the subtracted polynomial, writing -bx² - cx instead of -bx² + cx.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
(a - 4 + 2)x^3 + (-5 + b + 3)x^2 + (7 - c)x - 17 = 5x^3 - 6x^2 + 19x - 17
Correct
(a - 4 + 2)x^3 + (-5 - b + 3)x^2 + (7 + c)x - 17 = 5x^3 - 6x^2 + 19x - 17
When distributing the negative sign across the subtracted polynomial $-(4x^3 + bx^2 - cx + 9)$ , the signs of all terms inside must flip. The term $-(-cx)$ becomes $+cx$ , so the combined $x$ coefficient should be $7 + c$ rather than $7 - c$ .
Step 2
a - 2 = 5 $\implies$ a = 7
Step 3
-5 - b + 3 = -6 $\implies$ -b - 2 = -6 $\implies$ b = 4
Step 4
7 - c = 19 $\implies$ -c = 12 $\implies$ c = -12
Correct
7 + c = 19 $\implies$ c = 12
Your algebraic solving steps were executed properly, but the equation inherited the sign error from Step 1.
Step 5
a = 7, b = 4, c = -12
Correct
a = 7, b = 4, c = 12
Because of the upstream sign error in Step 1, the value of $c$ has the wrong sign.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly grouped like terms according to powers of $x$ .

You equated the corresponding polynomial coefficients systematically.

You accurately solved linear equations for $a$ and $b$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign distribution error across parentheses

You wrote $(7 - c)x$ instead of $(7 + c)x$ when subtracting $-cx$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You forgot that subtracting a negative term requires multiplying two negative signs, which results in a positive term: $-(-cx) = +cx$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute the negative sign completely: $-(A - B) = -A + B$
Write out the expanded terms explicitly with flipped signs before grouping coefficients.

CORRECTED PATH

ACEpath Teacher's Demonstration

Equate the correct $x$ coefficient to the target coefficient: $7 + c = 19$ .
Subtract $7$ from both sides: $c = 19 - 7 = 12$ .

Final answer

a = 7, b = 4, c = 12

NEXT STEPS

Final Takeaways from Your Teacher

Before combining like terms, write out an explicit step removing all parentheses by distributing $-1$ across every term inside.
Double-check double negatives: whenever you see $-(-...)$ , immediately circle it or write a bold $+$ sign above it.

OverviewACEpath Teacher's Diagnosis

02 The student misinterprets the subtracted term -(bx²) as simply keeping the plus sign when removing parentheses, leading to an incorrect sign for b.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
(a - 2)x^3 + (-5 + b + 3)x^2 + (7 + c)x - 17 = 5x^3 - 6x^2 + 19x - 17
Correct
(a - 2)x^3 + (-5 - b + 3)x^2 + (7 + c)x - 17 = 5x^3 - 6x^2 + 19x - 17
When subtracting $(4x^3 + bx^2 - cx + 9)$ , the leading minus sign distributes to every term inside the parentheses. The $x^2$ term should be $-(bx^2) = -bx^2$ , making the combined coefficient $(-5 - b + 3)$ rather than $(-5 + b + 3)$ .
Step 2
a - 2 = 5 $\implies$ a = 7
Step 3
b - 2 = -6 $\implies$ b = -4
Correct
-2 - b = -6 $\implies$ b = 4
Your arithmetic solving $b - 2 = -6$ was logically sound, but the equation itself inherited the missing negative sign from Step 1.
Step 4
7 + c = 19 $\implies$ c = 12
Step 5
a = 7, b = -4, c = 12
Correct
a = 7, b = 4, c = 12
The final value for $b$ is incorrect due to the sign error introduced in Step 1.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly distributed the negative sign to $-cx$ to obtain $+cx$ .

You grouped like terms systematically and equated corresponding polynomial coefficients.

You solved the linear equations for $a$ and $c$ accurately.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incomplete negative distribution

You forgot to distribute the leading subtraction sign to the second term $+bx^2$ , writing $+b$ instead of $-b$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating a subtracted polynomial group as if only the first term changes sign or selectively dropping the negative sign on interior positive terms.

REASONING PATH

Your ACEpath teacher’s suggested approach

Full Parentheses Expansion: $-(A + B - C + D) = -A - B + C - D$
Rewrite every term with its flipped sign in an expanded line before grouping coefficients together.

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand the subtracted polynomial carefully: $-(4x^3 + bx^2 - cx + 9) = -4x^3 - bx^2 + cx - 9$ .
Collect all $x^2$ terms: $-5x^2 - bx^2 + 3x^2 = (-5 - b + 3)x^2 = (-2 - b)x^2$ .
Equate with the target $x^2$ coefficient: $-2 - b = -6 \implies -b = -4 \implies b = 4$ .

Final answer

a = 7, b = 4, c = 12

NEXT STEPS

Final Takeaways from Your Teacher

Write out an explicit sign-expansion step before combining coefficients mentally.
Check each interior term after distributing: positive becomes negative, negative becomes positive.

OverviewACEpath Teacher's Diagnosis

03 The student adds rather than subtracts the coefficients of the second polynomial when grouping terms.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
(a+4+2)x^3 + (-5+b+3)x^2 + (7-c)x + 1 = 5x^3-6x^2+19x-17
Correct
(a - 4 + 2)x^3 + (-5 - b + 3)x^2 + (7 + c)x - 17 = 5x^3 - 6x^2 + 19x - 17
You omitted distributing the negative sign in front of the second set of parentheses, $-(4x^3 + bx^2 - cx + 9)$ . Every term inside that set of parentheses must reverse its sign to $-4x^3 - bx^2 + cx - 9$ .
Step 2
a + 6 = 5 $\implies$ a = -1 $\implies$ b - 2 = -6 $\implies$ b = -4 $\implies$ 7 - c = 19 $\implies$ c = -12
Correct
a - 2 = 5 $\implies$ a = 7 $\implies$ -2 - b = -6 $\implies$ b = 4 $\implies$ 7 + c = 19 $\implies$ c = 12
You correctly equated like-term coefficients with the right-hand side and solved each linear equation, but the equations used incorrect coefficient expressions from the previous sign distribution error.
Step 3
a = -1, b = -4, c = -12
Correct
a = 7, b = 4, c = 12
Your final constant values follow logically from your work above, but they are incorrect due to the initial distribution error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly recognized that simplifying polynomial expressions requires grouping like terms by degree.

You equated corresponding coefficients of identical powers between both sides of the equation.

You solved each resulting single-variable linear equation with accurate arithmetic.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign distribution error

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated the subtraction of a polynomial as simple addition or omitted distributing the factor of $-1$ to every individual term within the subtracted parentheses.

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute the negative sign fully: $-(p + q - r + s) = -p - q + r - s$
Always multiply each term inside parentheses by $-1$ and rewrite the expression without parentheses before combining like terms.
Double negatives become positive: -(-cx) = +cx
Watch for negative terms inside subtracted parentheses and explicitly convert them to additions.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute the negative sign across the middle polynomial: $-(4x^3 + bx^2 - cx + 9) = -4x^3 - bx^2 + cx - 9$ .
Group like terms: $(a - 4 + 2)x^3 + (-5 - b + 3)x^2 + (7 + c)x + (-9 - 8) = (a - 2)x^3 + (-2 - b)x^2 + (7 + c)x - 17$ .
Equate each coefficient to $5x^3 - 6x^2 + 19x - 17$ : $a - 2 = 5$ , $-2 - b = -6$ , and $7 + c = 19$ .

Final answer

a = 7, b = 4, c = 12

NEXT STEPS

Final Takeaways from Your Teacher

Rewrite polynomial subtraction by explicitly distributing $-1$ to every term inside the parentheses before combining terms.
Check each sign individually, paying special attention to double negatives like $-(-cx) = +cx$ .

Lock it in

Question

What is the result of removing parentheses in the expression $-(3x^2 - 5x + 4)$ ?

Answer

$-3x^2 + 5x - 4$ . Distribute the negative sign to every term inside: $-(-5x) = +5x$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to distribute the negative sign to the second and third terms inside the subtracted polynomial, writing -bx² - cx instead of -bx² + cx.

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 misinterprets the subtracted term -(bx²) as simply keeping the plus sign when removing parentheses, leading to an incorrect sign for b.

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 adds rather than subtracts the coefficients of the second polynomial when grouping terms.

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

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