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Answer :
Sure! Let's solve each of these equations step-by-step to find the solutions.
### Problem 19
Equation: [tex]\(-3x^3 - x^2 + 54x - 40 = 2x^2 + 6x + 20\)[/tex]
1. Rearrange all terms to one side:
[tex]\(-3x^3 - x^2 + 54x - 40 - 2x^2 - 6x - 20 = 0\)[/tex].
2. Combine like terms:
[tex]\(-3x^3 - 3x^2 + 48x - 60 = 0\)[/tex].
3. Factor or solve the polynomial equation:
The solutions to this equation are [tex]\(x = -5\)[/tex] and [tex]\(x = 2\)[/tex].
### Problem 20
Equation: [tex]\(2x^3 + 3x^2 - 36 = x^3 - x^2 + 9x\)[/tex]
1. Rearrange all terms to one side:
[tex]\(2x^3 + 3x^2 - 36 - x^3 + x^2 - 9x = 0\)[/tex].
2. Combine like terms:
[tex]\(x^3 + 4x^2 - 9x - 36 = 0\)[/tex].
3. Factor or solve the polynomial equation:
The solutions to this equation are [tex]\(x = -4\)[/tex], [tex]\(x = -3\)[/tex], and [tex]\(x = 3\)[/tex].
### Problem 21
Equation: [tex]\(-5x^4 + 4x^2 - 12x = -6x^4 + 3x^3\)[/tex]
1. Rearrange all terms to one side:
[tex]\(-5x^4 + 4x^2 - 12x + 6x^4 - 3x^3 = 0\)[/tex].
2. Combine like terms:
[tex]\(x^4 - 3x^3 + 4x^2 - 12x = 0\)[/tex].
3. Factor or solve the polynomial equation:
The solutions to this equation are [tex]\(x = 0\)[/tex], [tex]\(x = 3\)[/tex], [tex]\(x = -2i\)[/tex], and [tex]\(x = 2i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
These solutions represent the values of [tex]\(x\)[/tex] that satisfy each given equation!
### Problem 19
Equation: [tex]\(-3x^3 - x^2 + 54x - 40 = 2x^2 + 6x + 20\)[/tex]
1. Rearrange all terms to one side:
[tex]\(-3x^3 - x^2 + 54x - 40 - 2x^2 - 6x - 20 = 0\)[/tex].
2. Combine like terms:
[tex]\(-3x^3 - 3x^2 + 48x - 60 = 0\)[/tex].
3. Factor or solve the polynomial equation:
The solutions to this equation are [tex]\(x = -5\)[/tex] and [tex]\(x = 2\)[/tex].
### Problem 20
Equation: [tex]\(2x^3 + 3x^2 - 36 = x^3 - x^2 + 9x\)[/tex]
1. Rearrange all terms to one side:
[tex]\(2x^3 + 3x^2 - 36 - x^3 + x^2 - 9x = 0\)[/tex].
2. Combine like terms:
[tex]\(x^3 + 4x^2 - 9x - 36 = 0\)[/tex].
3. Factor or solve the polynomial equation:
The solutions to this equation are [tex]\(x = -4\)[/tex], [tex]\(x = -3\)[/tex], and [tex]\(x = 3\)[/tex].
### Problem 21
Equation: [tex]\(-5x^4 + 4x^2 - 12x = -6x^4 + 3x^3\)[/tex]
1. Rearrange all terms to one side:
[tex]\(-5x^4 + 4x^2 - 12x + 6x^4 - 3x^3 = 0\)[/tex].
2. Combine like terms:
[tex]\(x^4 - 3x^3 + 4x^2 - 12x = 0\)[/tex].
3. Factor or solve the polynomial equation:
The solutions to this equation are [tex]\(x = 0\)[/tex], [tex]\(x = 3\)[/tex], [tex]\(x = -2i\)[/tex], and [tex]\(x = 2i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
These solutions represent the values of [tex]\(x\)[/tex] that satisfy each given equation!
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