Answer :
Sure! Let's match each sum or difference with its simplified answer step by step.
We have the polynomials:
1. [tex]\( (5x^3 - 3x + 7) + (2x^3 + 6x^2 - x) \)[/tex]
2. [tex]\( (5x^3 - 3x + 7) - (2x^3 + 6x^2 - x) \)[/tex]
3. [tex]\( (7x^3 - 12x + 4) - (6x^2 - 8x - 3) \)[/tex]
4. [tex]\( (3x^3 + 10x^2 + 2x) - (10x^2 + 7x - 10) \)[/tex]
And the answers available:
1. [tex]\( 3x^3 - 5x + 10 \)[/tex]
2. [tex]\( 7x^3 - 6x^2 - 4x + 7 \)[/tex]
3. [tex]\( 3x^3 - 6x^2 - 2x + 7 \)[/tex]
4. [tex]\( 7x^3 + 6x^2 - 4x + 7 \)[/tex]
Step 1: Simplify each polynomial operation.
- For (1): [tex]\( (5x^3 - 3x + 7) + (2x^3 + 6x^2 - x) \)[/tex]
- Combine like terms:
- [tex]\(5x^3 + 2x^3 = 7x^3\)[/tex]
- [tex]\(6x^2\)[/tex] (no other [tex]\(x^2\)[/tex] term)
- [tex]\(-3x - x = -4x\)[/tex]
- [tex]\(+7\)[/tex] (constant term)
- Simplified form: [tex]\( 7x^3 + 6x^2 - 4x + 7 \)[/tex]
- For (2): [tex]\( (5x^3 - 3x + 7) - (2x^3 + 6x^2 - x) \)[/tex]
- Distribute the negative sign and combine like terms:
- [tex]\(5x^3 - 2x^3 = 3x^3\)[/tex]
- [tex]\(-6x^2\)[/tex] (subtracted [tex]\(6x^2\)[/tex] directly)
- [tex]\(-3x - (-x) = -3x + x = -2x\)[/tex]
- [tex]\(7\)[/tex] (constant term)
- Simplified form: [tex]\( 3x^3 - 6x^2 - 2x + 7 \)[/tex]
- For (3): [tex]\( (7x^3 - 12x + 4) - (6x^2 - 8x - 3) \)[/tex]
- Distribute the negative sign and combine like terms:
- [tex]\(7x^3\)[/tex] (no other [tex]\(x^3\)[/tex] term)
- [tex]\(-6x^2\)[/tex] (subtracted [tex]\(6x^2\)[/tex] directly)
- [tex]\(-12x - (-8x) = -12x + 8x = -4x\)[/tex]
- [tex]\(4 - (-3) = 4 + 3 = 7\)[/tex]
- Simplified form: [tex]\( 7x^3 - 6x^2 - 4x + 7 \)[/tex]
- For (4): [tex]\( (3x^3 + 10x^2 + 2x) - (10x^2 + 7x - 10) \)[/tex]
- Distribute the negative sign and combine like terms:
- [tex]\(3x^3\)[/tex] (no other [tex]\(x^3\)[/tex] term)
- [tex]\(10x^2 - 10x^2 = 0\)[/tex]
- [tex]\(2x - 7x = -5x\)[/tex]
- [tex]\(+10\)[/tex] (constant term)
- Simplified form: [tex]\( 3x^3 - 5x + 10 \)[/tex]
Step 2: Match each operation with the corresponding simplified answer:
- (1) matches with [tex]\( 7x^3 + 6x^2 - 4x + 7 \)[/tex] (Answer 2)
- (2) matches with [tex]\( 3x^3 - 6x^2 - 2x + 7 \)[/tex] (Answer 3)
- (3) matches with [tex]\( 7x^3 - 6x^2 - 4x + 7 \)[/tex] (Answer 4)
- (4) matches with [tex]\( 3x^3 - 5x + 10 \)[/tex] (Answer 1)
So, the matches are:
1. [tex]\( \left(5x^3-3x+7\right)+\left(2x^3+6x^2-x\right) \)[/tex] = Answer 2
2. [tex]\( \left(5x^3-3x+7\right)-\left(2x^3+6x^2-x\right) \)[/tex] = Answer 3
3. [tex]\( \left(7x^3-12x+4\right)-\left(6x^2-8x-3\right) \)[/tex] = Answer 4
4. [tex]\( \left(3x^3+10x^2+2x\right)-\left(10x^2+7x-10\right) \)[/tex] = Answer 1
We have the polynomials:
1. [tex]\( (5x^3 - 3x + 7) + (2x^3 + 6x^2 - x) \)[/tex]
2. [tex]\( (5x^3 - 3x + 7) - (2x^3 + 6x^2 - x) \)[/tex]
3. [tex]\( (7x^3 - 12x + 4) - (6x^2 - 8x - 3) \)[/tex]
4. [tex]\( (3x^3 + 10x^2 + 2x) - (10x^2 + 7x - 10) \)[/tex]
And the answers available:
1. [tex]\( 3x^3 - 5x + 10 \)[/tex]
2. [tex]\( 7x^3 - 6x^2 - 4x + 7 \)[/tex]
3. [tex]\( 3x^3 - 6x^2 - 2x + 7 \)[/tex]
4. [tex]\( 7x^3 + 6x^2 - 4x + 7 \)[/tex]
Step 1: Simplify each polynomial operation.
- For (1): [tex]\( (5x^3 - 3x + 7) + (2x^3 + 6x^2 - x) \)[/tex]
- Combine like terms:
- [tex]\(5x^3 + 2x^3 = 7x^3\)[/tex]
- [tex]\(6x^2\)[/tex] (no other [tex]\(x^2\)[/tex] term)
- [tex]\(-3x - x = -4x\)[/tex]
- [tex]\(+7\)[/tex] (constant term)
- Simplified form: [tex]\( 7x^3 + 6x^2 - 4x + 7 \)[/tex]
- For (2): [tex]\( (5x^3 - 3x + 7) - (2x^3 + 6x^2 - x) \)[/tex]
- Distribute the negative sign and combine like terms:
- [tex]\(5x^3 - 2x^3 = 3x^3\)[/tex]
- [tex]\(-6x^2\)[/tex] (subtracted [tex]\(6x^2\)[/tex] directly)
- [tex]\(-3x - (-x) = -3x + x = -2x\)[/tex]
- [tex]\(7\)[/tex] (constant term)
- Simplified form: [tex]\( 3x^3 - 6x^2 - 2x + 7 \)[/tex]
- For (3): [tex]\( (7x^3 - 12x + 4) - (6x^2 - 8x - 3) \)[/tex]
- Distribute the negative sign and combine like terms:
- [tex]\(7x^3\)[/tex] (no other [tex]\(x^3\)[/tex] term)
- [tex]\(-6x^2\)[/tex] (subtracted [tex]\(6x^2\)[/tex] directly)
- [tex]\(-12x - (-8x) = -12x + 8x = -4x\)[/tex]
- [tex]\(4 - (-3) = 4 + 3 = 7\)[/tex]
- Simplified form: [tex]\( 7x^3 - 6x^2 - 4x + 7 \)[/tex]
- For (4): [tex]\( (3x^3 + 10x^2 + 2x) - (10x^2 + 7x - 10) \)[/tex]
- Distribute the negative sign and combine like terms:
- [tex]\(3x^3\)[/tex] (no other [tex]\(x^3\)[/tex] term)
- [tex]\(10x^2 - 10x^2 = 0\)[/tex]
- [tex]\(2x - 7x = -5x\)[/tex]
- [tex]\(+10\)[/tex] (constant term)
- Simplified form: [tex]\( 3x^3 - 5x + 10 \)[/tex]
Step 2: Match each operation with the corresponding simplified answer:
- (1) matches with [tex]\( 7x^3 + 6x^2 - 4x + 7 \)[/tex] (Answer 2)
- (2) matches with [tex]\( 3x^3 - 6x^2 - 2x + 7 \)[/tex] (Answer 3)
- (3) matches with [tex]\( 7x^3 - 6x^2 - 4x + 7 \)[/tex] (Answer 4)
- (4) matches with [tex]\( 3x^3 - 5x + 10 \)[/tex] (Answer 1)
So, the matches are:
1. [tex]\( \left(5x^3-3x+7\right)+\left(2x^3+6x^2-x\right) \)[/tex] = Answer 2
2. [tex]\( \left(5x^3-3x+7\right)-\left(2x^3+6x^2-x\right) \)[/tex] = Answer 3
3. [tex]\( \left(7x^3-12x+4\right)-\left(6x^2-8x-3\right) \)[/tex] = Answer 4
4. [tex]\( \left(3x^3+10x^2+2x\right)-\left(10x^2+7x-10\right) \)[/tex] = Answer 1
