Answer :
Certainly! Let's simplify the given algebraic expressions by combining like terms. We need to group and reduce similar terms for each expression.
1) [tex]\(-6x^3 + 8x - 9x^2 + 18x^3 - 4x^2 + 6x\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-6x^3 + 18x^3 = 12x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-9x^2 - 4x^2 = -13x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(8x + 6x = 14x\)[/tex]
Simplified expression: [tex]\(12x^3 - 13x^2 + 14x\)[/tex]
2) [tex]\(20x^3 - 50x^4 + 18x^2 - 17x + 16x^4 - 9x^3 + 60x - 40x^2 + 30x\)[/tex]
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-50x^4 + 16x^4 = -34x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(20x^3 - 9x^3 = 11x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(18x^2 - 40x^2 = -22x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-17x + 60x + 30x = 73x\)[/tex]
Simplified expression: [tex]\(-34x^4 + 11x^3 - 22x^2 + 73x\)[/tex]
3) [tex]\(-60x^3 + 40x^2 - 20x + 90 + 20x^3 - 30x^2 + 12x - 50\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-60x^3 + 20x^3 = -40x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(40x^2 - 30x^2 = 10x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-20x + 12x = -8x\)[/tex]
- Combine the constant terms: [tex]\(90 - 50 = 40\)[/tex]
Simplified expression: [tex]\(-40x^3 + 10x^2 - 8x + 40\)[/tex]
4) [tex]\(16x^3 - 30x^2 + 40x - 60 - 40x^3 + 15x^2 - 17x + 30\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(16x^3 - 40x^3 = -24x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 15x^2 = -15x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(40x - 17x = 23x\)[/tex]
- Combine the constant terms: [tex]\(-60 + 30 = -30\)[/tex]
Simplified expression: [tex]\(-24x^3 - 15x^2 + 23x - 30\)[/tex]
5) [tex]\(-30x^4 + 20x^3 - 15x^2 + 30x - 50 + 50x^4 - 30x^3 + 10x^2 - 25x + 80\)[/tex]
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-30x^4 + 50x^4 = 20x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(20x^3 - 30x^3 = -10x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-15x^2 + 10x^2 = -5x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(30x - 25x = 5x\)[/tex]
- Combine the constant terms: [tex]\(-50 + 80 = 30\)[/tex]
Simplified expression: [tex]\(20x^4 - 10x^3 - 5x^2 + 5x + 30\)[/tex]
6) [tex]\(-50x^4 + 40x^3 - 25x^2 + 10x - 40 + 150x^4 - 25x^3 + 10x^2 - 30x + 60\)[/tex]
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-50x^4 + 150x^4 = 100x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(40x^3 - 25x^3 = 15x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-25x^2 + 10x^2 = -15x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(10x - 30x = -20x\)[/tex]
- Combine the constant terms: [tex]\(-40 + 60 = 20\)[/tex]
Simplified expression: [tex]\(100x^4 + 15x^3 - 15x^2 - 20x + 20\)[/tex]
7) [tex]\(-15x^3 + 16x^2 - 15x - 30 + 80x^3 - 50x^2 + 30x + 20\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-15x^3 + 80x^3 = 65x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(16x^2 - 50x^2 = -34x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-15x + 30x = 15x\)[/tex]
- Combine the constant terms: [tex]\(-30 + 20 = -10\)[/tex]
Simplified expression: [tex]\(65x^3 - 34x^2 + 15x - 10\)[/tex]
8) [tex]\(-60x^4 + 50x^3 - 60x^2 + 70x - 90 + 50x^4 - 30x^3 + 30x^2 - 35x + 50\)[/tex]
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-60x^4 + 50x^4 = -10x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(50x^3 - 30x^3 = 20x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-60x^2 + 30x^2 = -30x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(70x - 35x = 35x\)[/tex]
- Combine the constant terms: [tex]\(-90 + 50 = -40\)[/tex]
Simplified expression: [tex]\(-10x^4 + 20x^3 - 30x^2 + 35x - 40\)[/tex]
9) [tex]\(-25x^3 + 50x^2 - 15x + 30 + 40x^3 - 35x^2 + 35x - 50\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-25x^3 + 40x^3 = 15x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(50x^2 - 35x^2 = 15x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-15x + 35x = 20x\)[/tex]
- Combine the constant terms: [tex]\(30 - 50 = -20\)[/tex]
Simplified expression: [tex]\(15x^3 + 15x^2 + 20x - 20\)[/tex]
10) [tex]\(-6x^3 - 15x^2 + 8x - 16 + 25x^3 + 25x^2 + 7x + 30\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-6x^3 + 25x^3 = 19x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-15x^2 + 25x^2 = 10x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(8x + 7x = 15x\)[/tex]
- Combine the constant terms: [tex]\(-16 + 30 = 14\)[/tex]
Simplified expression: [tex]\(19x^3 + 10x^2 + 15x + 14\)[/tex]
Hopefully, these step-by-step simplifications make the process clear! If you have any questions or need further explanation, feel free to ask.
1) [tex]\(-6x^3 + 8x - 9x^2 + 18x^3 - 4x^2 + 6x\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-6x^3 + 18x^3 = 12x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-9x^2 - 4x^2 = -13x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(8x + 6x = 14x\)[/tex]
Simplified expression: [tex]\(12x^3 - 13x^2 + 14x\)[/tex]
2) [tex]\(20x^3 - 50x^4 + 18x^2 - 17x + 16x^4 - 9x^3 + 60x - 40x^2 + 30x\)[/tex]
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-50x^4 + 16x^4 = -34x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(20x^3 - 9x^3 = 11x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(18x^2 - 40x^2 = -22x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-17x + 60x + 30x = 73x\)[/tex]
Simplified expression: [tex]\(-34x^4 + 11x^3 - 22x^2 + 73x\)[/tex]
3) [tex]\(-60x^3 + 40x^2 - 20x + 90 + 20x^3 - 30x^2 + 12x - 50\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-60x^3 + 20x^3 = -40x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(40x^2 - 30x^2 = 10x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-20x + 12x = -8x\)[/tex]
- Combine the constant terms: [tex]\(90 - 50 = 40\)[/tex]
Simplified expression: [tex]\(-40x^3 + 10x^2 - 8x + 40\)[/tex]
4) [tex]\(16x^3 - 30x^2 + 40x - 60 - 40x^3 + 15x^2 - 17x + 30\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(16x^3 - 40x^3 = -24x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 15x^2 = -15x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(40x - 17x = 23x\)[/tex]
- Combine the constant terms: [tex]\(-60 + 30 = -30\)[/tex]
Simplified expression: [tex]\(-24x^3 - 15x^2 + 23x - 30\)[/tex]
5) [tex]\(-30x^4 + 20x^3 - 15x^2 + 30x - 50 + 50x^4 - 30x^3 + 10x^2 - 25x + 80\)[/tex]
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-30x^4 + 50x^4 = 20x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(20x^3 - 30x^3 = -10x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-15x^2 + 10x^2 = -5x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(30x - 25x = 5x\)[/tex]
- Combine the constant terms: [tex]\(-50 + 80 = 30\)[/tex]
Simplified expression: [tex]\(20x^4 - 10x^3 - 5x^2 + 5x + 30\)[/tex]
6) [tex]\(-50x^4 + 40x^3 - 25x^2 + 10x - 40 + 150x^4 - 25x^3 + 10x^2 - 30x + 60\)[/tex]
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-50x^4 + 150x^4 = 100x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(40x^3 - 25x^3 = 15x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-25x^2 + 10x^2 = -15x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(10x - 30x = -20x\)[/tex]
- Combine the constant terms: [tex]\(-40 + 60 = 20\)[/tex]
Simplified expression: [tex]\(100x^4 + 15x^3 - 15x^2 - 20x + 20\)[/tex]
7) [tex]\(-15x^3 + 16x^2 - 15x - 30 + 80x^3 - 50x^2 + 30x + 20\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-15x^3 + 80x^3 = 65x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(16x^2 - 50x^2 = -34x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-15x + 30x = 15x\)[/tex]
- Combine the constant terms: [tex]\(-30 + 20 = -10\)[/tex]
Simplified expression: [tex]\(65x^3 - 34x^2 + 15x - 10\)[/tex]
8) [tex]\(-60x^4 + 50x^3 - 60x^2 + 70x - 90 + 50x^4 - 30x^3 + 30x^2 - 35x + 50\)[/tex]
- Combine the [tex]\(x^4\)[/tex] terms: [tex]\(-60x^4 + 50x^4 = -10x^4\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(50x^3 - 30x^3 = 20x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-60x^2 + 30x^2 = -30x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(70x - 35x = 35x\)[/tex]
- Combine the constant terms: [tex]\(-90 + 50 = -40\)[/tex]
Simplified expression: [tex]\(-10x^4 + 20x^3 - 30x^2 + 35x - 40\)[/tex]
9) [tex]\(-25x^3 + 50x^2 - 15x + 30 + 40x^3 - 35x^2 + 35x - 50\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-25x^3 + 40x^3 = 15x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(50x^2 - 35x^2 = 15x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-15x + 35x = 20x\)[/tex]
- Combine the constant terms: [tex]\(30 - 50 = -20\)[/tex]
Simplified expression: [tex]\(15x^3 + 15x^2 + 20x - 20\)[/tex]
10) [tex]\(-6x^3 - 15x^2 + 8x - 16 + 25x^3 + 25x^2 + 7x + 30\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-6x^3 + 25x^3 = 19x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-15x^2 + 25x^2 = 10x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(8x + 7x = 15x\)[/tex]
- Combine the constant terms: [tex]\(-16 + 30 = 14\)[/tex]
Simplified expression: [tex]\(19x^3 + 10x^2 + 15x + 14\)[/tex]
Hopefully, these step-by-step simplifications make the process clear! If you have any questions or need further explanation, feel free to ask.
