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Perform the indicated polynomial operations and express the result in standard form: (3x3−5x2+2x−7)−(x3+4x2−6x+9)+2x(2x2−3x+1)(3x^3 - 5x^2 + 2x - 7) - (x^3 + 4x^2 - 6x + 9) + 2x(2x^2 - 3x + 1).

Answer and steps

Final answer

6x3−15x2+10x−166x^3 - 15x^2 + 10x - 16

Steps

01

Remove parentheses and distribute

To simplify the expression and write it in standard form, follow the order of operations by expanding each part and then combining like terms.

  • The first group remains unchanged: 3x3−5x2+2x−73x^3 - 5x^2 + 2x - 7
  • Distribute the negative sign across every term inside the second set of parentheses: −(x3+4x2−6x+9)=−x3−4x2+6x−9-(x^3 + 4x^2 - 6x + 9) = -x^3 - 4x^2 + 6x - 9 (Common pitfall: Ensure the sign of every term changes, particularly turning −6x-6x into +6x+6x and +9+9 into −9-9.)
  • Use the distributive property to multiply the monomial 2x2x by each term inside the third set of parentheses: 2x(2x2−3x+1)=(2x⋅2x2)+(2x⋅−3x)+(2x⋅1)=4x3−6x2+2x2x(2x^2 - 3x + 1) = (2x \cdot 2x^2) + (2x \cdot -3x) + (2x \cdot 1) = 4x^3 - 6x^2 + 2x
02

Write all expanded terms together

Combine the expanded components into a single expression: (3x3−5x2+2x−7)+(−x3−4x2+6x−9)+(4x3−6x2+2x)(3x^3 - 5x^2 + 2x - 7) + (-x^3 - 4x^2 + 6x - 9) + (4x^3 - 6x^2 + 2x)

03

Group and combine like terms

Group terms that have the same variable and exponent:

  • x3x^3 terms: 3x3−x3+4x3=(3−1+4)x3=6x33x^3 - x^3 + 4x^3 = (3 - 1 + 4)x^3 = 6x^3
  • x2x^2 terms: −5x2−4x2−6x2=(−5−4−6)x2=−15x2-5x^2 - 4x^2 - 6x^2 = (-5 - 4 - 6)x^2 = -15x^2
  • xx terms: 2x+6x+2x=(2+6+2)x=10x2x + 6x + 2x = (2 + 6 + 2)x = 10x
  • Constant terms: −7−9=−16-7 - 9 = -16
04

Write in standard form

Arrange the terms in descending order of their exponents: 6x3−15x2+10x−166x^3 - 15x^2 + 10x - 16

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student only applies the subtraction to the first term of the second polynomial and fails to distribute the negative sign to the remaining terms, yielding 6x³ - 7x² - 2x - 2.

Sample Student Work

Reviewed by ACEpath Teacher
Step-by-Step Review
Walk Through It with Your Teacher
Incorrect
  1. Step 1

    = 3x^3 - 5x^2 + 2x - 7 - x^3 + 4x^2 - 6x + 9 + 4x^3 - 6x^2 + 2x

    Correct

    3x^3 - 5x^2 + 2x - 7 - x^3 - 4x^2 + 6x - 9 + 4x^3 - 6x^2 + 2x

    When subtracting a polynomial, the negative sign must be distributed to every term inside the parentheses: −(x3+4x2−6x+9)=−x3−4x2+6x−9-(x^3 + 4x^2 - 6x + 9) = -x^3 - 4x^2 + 6x - 9. You only applied the negative sign to the first term x3x^3.

  2. Step 2

    = (3 - 1 + 4)x^3 + (-5 + 4 - 6)x^2 + (2 - 6 + 2)x + (-7 + 9)

    Correct

    (3 - 1 + 4)x^3 + (-5 - 4 - 6)x^2 + (2 + 6 + 2)x + (-7 - 9)

    Your grouping method was logically sound, but the coefficients for x2x^2, xx, and the constant term inherited incorrect signs from the previous step.

  3. Step 3

    = 6x^3 - 7x^2 - 2x + 2

    Correct

    6x^3 - 15x^2 + 10x - 16

    The final arithmetic followed your grouped coefficients correctly, but the result differs from the correct standard form due to the earlier sign error.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly multiplied 2x2x through the third polynomial using the distributive property.

You correctly organized and collected like terms by grouping coefficients of matching powers.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Distributive sign error in polynomial subtraction

You treated the subtraction sign in front of the second polynomial as applying only to the leading term rather than distributing −1-1 to every term inside the parentheses.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Failing to treat a preceding minus sign as a multiplier of −1-1 that flips the sign of every individual term inside the grouping symbols.

REASONING PATH
Your ACEpath teacher’s suggested approach
Distribute the negative sign

  • −(a+b−c)=−a−b+c-(a + b - c) = -a - b + c

  • Rewrite subtraction of a polynomial as multiplying each term inside the parentheses by −1-1 before removing the parentheses.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Distribute −1-1 across all four terms of the second polynomial: −(x3+4x2−6x+9)=−x3−4x2+6x−9-(x^3 + 4x^2 - 6x + 9) = -x^3 - 4x^2 + 6x - 9.

  2. Combine like terms with the correct signs: (3−1+4)x3+(−5−4−6)x2+(2+6+2)x+(−7−9)(3 - 1 + 4)x^3 + (-5 - 4 - 6)x^2 + (2 + 6 + 2)x + (-7 - 9).

Final answer

6x^3 - 15x^2 + 10x - 16

NEXT STEPS
Final Takeaways from Your Teacher

  • Before combining like terms, explicitly write out +(−1)(… )+(-1)(\dots) or draw arrows to every term inside the subtracted polynomial to verify that every sign has reversed.

  • Practice distributing negative signs across polynomials with mixed positive and negative coefficients.

OverviewACEpath Teacher's Diagnosis

02 The student multiplies the coefficients but fails to add exponents when multiplying variables, writing 2x(2x²) as 4x² instead of 4x³ and obtaining 2x³ - 11x² + 10x - 16.

Sample Student Work

Reviewed by ACEpath Teacher
Step-by-Step Review
Walk Through It with Your Teacher
Incorrect
  1. Step 1

    = 3x^3 - 5x^2 + 2x - 7 - x^3 - 4x^2 + 6x - 9 + 4x^2 - 6x + 2x

    Correct

    3x^3 - 5x^2 + 2x - 7 - x^3 - 4x^2 + 6x - 9 + 4x^3 - 6x^2 + 2x

    When expanding 2x(2x2−3x+1)2x(2x^2 - 3x + 1), remember that x=x1x = x^1. Multiplying by xx increases each exponent by 11: 2x⋅2x2=4x32x \cdot 2x^2 = 4x^3 and 2x⋅(−3x)=−6x22x \cdot (-3x) = -6x^2.

  2. Step 2

    = (3-1)x^3 + (-5-4+4)x^2 + (2+6-6+2)x + (-7-9)

    Correct

    (3 - 1 + 4)x^3 + (-5 - 4 - 6)x^2 + (2 + 6 + 2)x + (-7 - 9)

    You grouped your terms systematically, but the coefficients reflect the earlier exponent error in the expansion.

  3. Step 3

    = 2x^3 - 5x^2 + 4x - 16

    Correct

    6x^3 - 15x^2 + 10x - 16

    The final arithmetic followed your previous step correctly, but evaluates to an incorrect polynomial due to the earlier expansion error.

WHAT YOU DID WELL
Compliments from Your Teacher

You distributed the negative sign across −(x3+4x2−6x+9)-(x^3 + 4x^2 - 6x + 9) completely and accurately.

You organized and combined like terms systematically using coefficient grouping.

You performed all integer additions and subtractions correctly.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Exponent addition omission in monomial multiplication

You treated multiplying by 2x2x as if you were multiplying by 22, forgetting to add the variable power x1x^1 to each existing term.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Missing the exponent rule xm⋅xn=xm+nx^m \cdot x^n = x^{m+n} during monomial distribution over polynomials, specifically forgetting that a variable with no written exponent has a power of 11 (x=x1x = x^1).

REASONING PATH
Your ACEpath teacher’s suggested approach
Make the hidden exponent visible

  • Every variable with no exponent has an implicit power of 11: x=x1x = x^1.

  • Write 2x12x^1 before distributing so you remember to add 11 to every variable's exponent.

Product rule for multiplying terms

  • (a x^m)(b x^n) = (a ⋅\cdot b)x^{m+n}$.

  • Multiply the numerical coefficients, and add the powers of matching variables.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Expand the third group correctly: 2x(2x2−3x+1)=4x3−6x2+2x2x(2x^2 - 3x + 1) = 4x^3 - 6x^2 + 2x.

  2. Combine with the other two expanded groups: (3x3−x3+4x3)+(−5x2−4x2−6x2)+(2x+6x+2x)+(−7−9)(3x^3 - x^3 + 4x^3) + (-5x^2 - 4x^2 - 6x^2) + (2x + 6x + 2x) + (-7 - 9).

  3. Simplify each coefficient: 6x3−15x2+10x−166x^3 - 15x^2 + 10x - 16.

Final answer

6x3−15x2+10x−166x^3 - 15x^2 + 10x - 16

NEXT STEPS
Final Takeaways from Your Teacher

  • Practice distributing monomials with variables like 3x(x2−4x+2)3x(x^2 - 4x + 2) to build automatic exponent addition.

  • Whenever distributing a variable term, annotate the exponent 11 above the variable before multiplying.

OverviewACEpath Teacher's Diagnosis

03 The student adds coefficients with opposite signs incorrectly when combining like terms for the x² components, calculating -5 - 4 - 6 as -3 and obtaining 6x³ - 3x² + 10x - 16.

Sample Student Work

Reviewed by ACEpath Teacher
Step-by-Step Review
Walk Through It with Your Teacher
Partially correct
  1. Step 1

    = 3x^3 - 5x^2 + 2x - 7 - x^3 - 4x^2 + 6x - 9 + 4x^3 - 6x^2 + 2x

  2. Step 2

    = (3 - 1 + 4)x^3 + (-5 - 4 - 6)x^2 + (2 + 6 + 2)x + (-7 - 9)

  3. Step 3

    = 6x^3 - 3x^2 + 10x - 16

    Correct

    6x^3 - 15x^2 + 10x - 16

    When evaluating the x2x^2 coefficient, (−5−4−6)(-5 - 4 - 6) equals −15-15, not −3-3.

WHAT YOU DID WELL
Compliments from Your Teacher

You distributed the negative sign to every term inside −(x3+4x2−6x+9)-(x^3 + 4x^2 - 6x + 9) without missing any sign flips.

You accurately distributed the monomial 2x2x across (2x2−3x+1)(2x^2 - 3x + 1).

You grouped like terms cleanly and accurately computed the coefficients for x3x^3, xx, and the constant term.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Arithmetic with Negative Numbers

You calculated −5−4−6-5 - 4 - 6 as −3-3 instead of −15-15.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Accumulating multiple negative integers can lead to accidentally subtracting magnitudes (e.g., treating −9−6-9 - 6 as −9+6=−3-9 + 6 = -3) rather than continuing to add their absolute values in the negative direction.

REASONING PATH
Your ACEpath teacher’s suggested approach
Adding consecutive negatives

  • −a−b−c=−(a+b+c)-a - b - c = -(a + b + c)

  • Factor out the negative sign mentally or on scratch paper: −5−4−6=−(5+4+6)=−15-5 - 4 - 6 = -(5 + 4 + 6) = -15.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Evaluate the x3x^3 coefficient: 3−1+4=63 - 1 + 4 = 6

  2. Evaluate the x2x^2 coefficient: −5−4−6=−9−6=−15-5 - 4 - 6 = -9 - 6 = -15

  3. Evaluate the xx coefficient: 2+6+2=102 + 6 + 2 = 10

  4. Evaluate the constant: −7−9=−16-7 - 9 = -16

Final answer

6x3−15x2+10x−166x^3 - 15x^2 + 10x - 16

NEXT STEPS
Final Takeaways from Your Teacher

  • Whenever you combine three or more negative coefficients, factor out a negative sign: (−a−b−c)=−(a+b+c)(-a - b - c) = -(a + b + c).

  • Take an extra two seconds to verify negative sums sequentially: −5−4=−9-5 - 4 = -9, then −9−6=−15-9 - 6 = -15.

Lock it in

Question

What does a minus sign directly in front of parentheses, like −(a−b+c)-(a - b + c), do to each term inside?

Answer

It multiplies every term by −1-1, reversing every sign: −a+b−c-a + b - c.

Similar question

01

Simplify the expression and write the result in standard form: (4x2−3x+5)−(2x2+5x−8)(4x^2 - 3x + 5) - (2x^2 + 5x - 8).

Answer

2x2−8x+132x^2 - 8x + 13. Distribute the negative sign: 4x2−3x+5−2x2−5x+84x^2 - 3x + 5 - 2x^2 - 5x + 8. Combine like terms: (4−2)x2+(−3−5)x+(5+8)=2x2−8x+13(4 - 2)x^2 + (-3 - 5)x + (5 + 8) = 2x^2 - 8x + 13.

02

Simplify and express in standard form: 3(x2−4x+2)−(5x2−2x−7)+4x(x−1)3(x^2 - 4x + 2) - (5x^2 - 2x - 7) + 4x(x - 1).

Answer

2x2−14x+132x^2 - 14x + 13. Expand each term: 3(x2−4x+2)=3x2−12x+63(x^2 - 4x + 2) = 3x^2 - 12x + 6, −(5x2−2x−7)=−5x2+2x+7-(5x^2 - 2x - 7) = -5x^2 + 2x + 7, and 4x(x−1)=4x2−4x4x(x - 1) = 4x^2 - 4x. Combine like terms: (3−5+4)x2+(−12+2−4)x+(6+7)=2x2−14x+13(3 - 5 + 4)x^2 + (-12 + 2 - 4)x + (6 + 7) = 2x^2 - 14x + 13.

03

Subtract (2x3−6x2+3x−11)(2x^3 - 6x^2 + 3x - 11) from the sum of (x3+5x2−4)(x^3 + 5x^2 - 4) and 3x(x2−2x+5)3x(x^2 - 2x + 5), expressing your answer in standard form.

Answer

2x3+5x2+12x+72x^3 + 5x^2 + 12x + 7. First sum the two polynomials: (x3+5x2−4)+(3x3−6x2+15x)=4x3−x2+15x−4(x^3 + 5x^2 - 4) + (3x^3 - 6x^2 + 15x) = 4x^3 - x^2 + 15x - 4. Then subtract the third polynomial: (4x3−x2+15x−4)−(2x3−6x2+3x−11)=4x3−x2+15x−4−2x3+6x2−3x+11(4x^3 - x^2 + 15x - 4) - (2x^3 - 6x^2 + 3x - 11) = 4x^3 - x^2 + 15x - 4 - 2x^3 + 6x^2 - 3x + 11. Group like terms: (4−2)x3+(−1+6)x2+(15−3)x+(−4+11)=2x3+5x2+12x+7(4 - 2)x^3 + (-1 + 6)x^2 + (15 - 3)x + (-4 + 11) = 2x^3 + 5x^2 + 12x + 7.

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