An unknown polynomial $P(x)$ satisfies the equation $P(x) - (4x^3 - 7x^2 + 5x - 12) = (2x - 3)(x^2 + 4x - 5)$. Solve backwards for the original polynomial $P(x)$ and write it in standard form.

An unknown polynomial $P(x)$ satisfies the equation $P(x) - (4x^3 - 7x^2 + 5x - 12) = (2x - 3)(x^2 + 4x - 5)$ . Solve backwards for the original polynomial $P(x)$ and write it in standard form.

Final answer

$P(x) = 6x^3 - 2x^2 - 17x + 3$

Steps

Expand the product on the right-hand side

To solve backwards for $P(x)$ , we can isolate $P(x)$ on one side of the equation by expanding the product on the right-hand side and then adding the subtracted polynomial to both sides.

Use the distributive property to multiply the binomial by the trinomial: $(2x - 3)(x^2 + 4x - 5) = 2x(x^2 + 4x - 5) - 3(x^2 + 4x - 5)$

Distribute each term: $2x(x^2 + 4x - 5) = 2x^3 + 8x^2 - 10x$ $-3(x^2 + 4x - 5) = -3x^2 - 12x + 15$

Combine like terms: $(2x^3 + 8x^2 - 10x) + (-3x^2 - 12x + 15) = 2x^3 + (8x^2 - 3x^2) + (-10x - 12x) + 15$ $= 2x^3 + 5x^2 - 22x + 15$

Isolate $P(x)$

Substitute this expanded expression back into the original equation: $P(x) - (4x^3 - 7x^2 + 5x - 12) = 2x^3 + 5x^2 - 22x + 15$

Add $(4x^3 - 7x^2 + 5x - 12)$ to both sides: $P(x) = (2x^3 + 5x^2 - 22x + 15) + (4x^3 - 7x^2 + 5x - 12)$

Combine like terms

Group terms by their degree:

$x^3$ terms: $2x^3 + 4x^3 = 6x^3$
$x^2$ terms: $5x^2 + (-7x^2) = -2x^2$
$x$ terms: $-22x + 5x = -17x$
Constant terms: $15 + (-12) = 3$

Writing the polynomial in standard form (from highest to lowest degree): $P(x) = 6x^3 - 2x^2 - 17x + 3$

Common Pitfall: When distributing the negative term $-3$ across $(x^2 + 4x - 5)$ , take care to reverse the signs of each term inside the parentheses (e.g., $-3 \cdot (-5) = +15$ , not $-15$ ).

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student subtracts the polynomial (4x³ - 7x² + 5x - 12) from the right-hand side instead of adding it when isolating P(x).

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
(2x - 3)(x^2 + 4x - 5) = 2x^3 + 8x^2 - 10x - 3x^2 - 12x + 15 ⟹ 2x^3 + 5x^2 - 22x + 15
Step 2
P(x) = (2x^3 + 5x^2 - 22x + 15) - (4x^3 - 7x^2 + 5x - 12)
Correct
P(x) = (2x^3 + 5x^2 - 22x + 15) + (4x^3 - 7x^2 + 5x - 12)
Because the given equation is $P(x) - (4x^3 - 7x^2 + 5x - 12) = R(x)$ , the inverse operation to isolate $P(x)$ is adding $(4x^3 - 7x^2 + 5x - 12)$ to both sides, not subtracting it.
Step 3
P(x) = -2x^3 + 12x^2 - 27x + 27
Correct
P(x) = 6x^3 - 2x^2 - 17x + 3
This result follows your arithmetic from Step 2, but it inherits the sign error made when isolating $P(x)$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded the product of the binomial and trinomial completely and correctly.

You distributed the negative sign and combined like terms accurately according to your written equation.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Inverse Operation Error

You wrote a subtraction sign instead of an addition sign when moving the polynomial to the other side.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the inverse operation required to isolate an unknown variable when an entire grouped expression is subtracted.

REASONING PATH

Your ACEpath teacher’s suggested approach

Identify the connecting operation: If an expression has the form $A - B = C$ , then $A = C + B$ .
Check the sign immediately in front of the group being moved across the equals sign and apply its inverse.

CORRECTED PATH

ACEpath Teacher's Demonstration

Write the equation with the correct inverse operation: $P(x) = (2x^3 + 5x^2 - 22x + 15) + (4x^3 - 7x^2 + 5x - 12)$ .
Combine like terms by degree: $(2 + 4)x^3 + (5 - 7)x^2 + (-22 + 5)x + (15 - 12)$ .

Final answer

P(x) = 6x^3 - 2x^2 - 17x + 3

NEXT STEPS

Final Takeaways from Your Teacher

Before combining terms, explicitly write $+ (\text{subtracted polynomial})$ on both sides of the equation to ensure the inverse operation is applied correctly.
Perform a quick mental check: if $P - B = C$ , then $P$ must equal $C + B$ .

OverviewACEpath Teacher's Diagnosis

02 The student fails to distribute the negative sign to the constant term when multiplying -3 by -5, writing -15 instead of +15.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$(2x - 3)(x^2 + 4x - 5) = 2x^3 + 8x^2 - 10x - 3x^2 - 12x - 15 \implies = 2x^3 + 5x^2 - 22x - 15$
Correct
$(2x - 3)(x^2 + 4x - 5) = 2x^3 + 8x^2 - 10x - 3x^2 - 12x + 15 = 2x^3 + 5x^2 - 22x + 15$
When distributing $-3$ across $(x^2 + 4x - 5)$ , multiplying two negative numbers gives a positive product: $(-3) \cdot (-5) = +15$ , not $-15$ .
Step 2
$P(x) = (2x^3 + 5x^2 - 22x - 15) + (4x^3 - 7x^2 + 5x - 12)$
Correct
$P(x) = (2x^3 + 5x^2 - 22x + 15) + (4x^3 - 7x^2 + 5x - 12)$
Your algebraic setup to isolate $P(x)$ by adding the subtracted polynomial is completely correct, but it inherits the $-15$ constant from the previous step.
Step 3
$P(x) = 6x^3 - 2x^2 - 17x - 27$
Correct
$P(x) = 6x^3 - 2x^2 - 17x + 3$
You accurately combined like terms across all polynomial degrees, but due to the inherited constant error, $-15 + (-12)$ produced $-27$ instead of $15 + (-12) = 3$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly recognized that adding the subtracted polynomial isolates $P(x)$ .

You accurately distributed $2x$ and the variable terms of $-3$ across the trinomial.

You combined all like terms accurately according to their degrees.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign distribution error with negative factors

Multiplying the negative factor $-3$ by the negative constant $-5$ resulted in $-15$ rather than $+15$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking the double negative rule when expanding brackets, causing the negative sign in front of the multiplier not to flip the sign of a subtracted term inside.

REASONING PATH

Your ACEpath teacher’s suggested approach

Double Negative Product Rule: The product of two negative numbers is always positive: $(-a) \cdot (-b) = +(ab)$ .
Circle or explicitly include the sign with each term before distributing: $(-3) \times (-5) = +15$ .
Bracket Expansion Check: Multiplying by a negative monomial reverses the sign of every single term inside the parentheses.
Verify that a positive term inside becomes negative and a negative term inside becomes positive.

CORRECTED PATH

ACEpath Teacher's Demonstration

Correct the expansion of $(2x - 3)(x^2 + 4x - 5)$ : $2x^3 + 8x^2 - 10x - 3x^2 - 12x + 15 = 2x^3 + 5x^2 - 22x + 15$ .
Substitute the corrected polynomial into the isolated equation: $P(x) = (2x^3 + 5x^2 - 22x + 15) + (4x^3 - 7x^2 + 5x - 12)$ .
Combine like terms: $(2+4)x^3 + (5-7)x^2 + (-22+5)x + (15-12) = 6x^3 - 2x^2 - 17x + 3$ .

Final answer

$P(x) = 6x^3 - 2x^2 - 17x + 3$

NEXT STEPS

Final Takeaways from Your Teacher

Practice distributing negative numbers across expressions containing negative terms to reinforce sign flipping.
Highlight or circle signs along with their terms before carrying out polynomial multiplication.
Perform a quick mental constant-term check: $(-3) \times (-5) + (-12) = 15 - 12 = 3$ .

OverviewACEpath Teacher's Diagnosis

03 The student incorrectly adds the linear terms during polynomial expansion, computing -10x - 12x as -2x instead of -22x.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
(2x - 3)(x^2 + 4x - 5) = 2x^3 + 8x^2 - 10x - 3x^2 - 12x + 15
Step 2
= 2x^3 + 5x^2 - 2x + 15
Correct
= 2x^3 + 5x^2 - 22x + 15
When combining the linear terms $-10x - 12x$ , you calculated $-2x$ instead of $-22x$ . Both coefficients are negative, so their magnitudes add together: $-10 - 12 = -22$ .
Step 3
P(x) = (2x^3 + 5x^2 - 2x + 15) + (4x^3 - 7x^2 + 5x - 12)
Correct
P(x) = (2x^3 + 5x^2 - 22x + 15) + (4x^3 - 7x^2 + 5x - 12)
You correctly isolated $P(x)$ by adding $(4x^3 - 7x^2 + 5x - 12)$ to both sides, though this step inherits the $-2x$ term from the previous step.
Step 4
P(x) = 6x^3 - 2x^2 + 3x + 3
Correct
P(x) = 6x^3 - 2x^2 - 17x + 3
Your final addition was carried out accurately based on your intermediate expression (since $-2x + 5x = 3x$ ), but the final linear term is incorrect due to the propagated error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly applied the distributive property to expand the product $(2x - 3)(x^2 + 4x - 5)$ .

You correctly identified that isolating $P(x)$ requires adding the subtracted polynomial to the right-hand side.

You accurately combined all other like terms ( $x^3$ , $x^2$ , and constant terms) throughout both stages of the problem.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Combining Negative Coefficients

You combined $-10x - 12x$ as $-2x$ instead of $-22x$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

When combining two terms with negative signs, you subtracted their absolute values (treating $-10 - 12$ like $-12 + 10$ or $12 - 10$ ) instead of summing their magnitudes and retaining the negative sign.

REASONING PATH

Your ACEpath teacher’s suggested approach

Same-Sign Addition Rule: When adding two numbers with the same sign, add their absolute values and keep the common sign: $-a - b = -(a + b)$ .
Whenever you see two negative terms being collected, rewrite the coefficients as $-(10 + 12)x = -22x$ before simplifying.

CORRECTED PATH

ACEpath Teacher's Demonstration

Combine the linear terms correctly: $-10x - 12x = -(10 + 12)x = -22x$ , giving the expanded product $2x^3 + 5x^2 - 22x + 15$ .
Substitute this product into the isolated equation: $P(x) = (2x^3 + 5x^2 - 22x + 15) + (4x^3 - 7x^2 + 5x - 12)$ .
Combine like terms by degree: $x^3$ terms give $2 + 4 = 6$ , $x^2$ terms give $5 - 7 = -2$ , $x$ terms give $-22 + 5 = -17$ , and constants give $15 - 12 = 3$ .

Final answer

P(x) = 6x^3 - 2x^2 - 17x + 3

NEXT STEPS

Final Takeaways from Your Teacher

Practice combining pairs of negative integers on a quick number-line drill to reinforce $-a - b = -(a + b)$ .
When expanding polynomials with several terms, underline or group like terms with their preceding signs before combining them.

Lock it in

Question

If $A - B = C$ , how do you isolate $A$ using an inverse operation?

Answer

$A = C + B$ . Addition is the inverse of subtraction, so add $B$ to both sides.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student subtracts the polynomial (4x³ - 7x² + 5x - 12) from the right-hand side instead of adding it when isolating P(x).

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 fails to distribute the negative sign to the constant term when multiplying -3 by -5, writing -15 instead of +15.

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 incorrectly adds the linear terms during polynomial expansion, computing -10x - 12x as -2x instead of -22x.

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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