Answer :
Let's approach these polynomial division problems step-by-step.
### Problem 1:
Dividing [tex]\( 3x^3 + 4x^2 - 5x + 2 \)[/tex] by [tex]\( x - 2 \)[/tex]
1. Set up the division: To divide [tex]\( 3x^3 + 4x^2 - 5x + 2 \)[/tex] by [tex]\( x - 2 \)[/tex], we use polynomial long division.
2. Divide the leading term:
- Divide the leading term of the numerator [tex]\( 3x^3 \)[/tex] by the leading term of the denominator [tex]\( x \)[/tex]:
[tex]\[ \frac{3x^3}{x} = 3x^2 \][/tex]
3. Multiply and subtract:
- Multiply [tex]\( 3x^2 \)[/tex] by [tex]\( x - 2 \)[/tex]:
[tex]\[ 3x^2 \cdot (x - 2) = 3x^3 - 6x^2 \][/tex]
- Subtract this from the original polynomial:
[tex]\[ (3x^3 + 4x^2 - 5x + 2) - (3x^3 - 6x^2) = 10x^2 - 5x + 2 \][/tex]
4. Repeat the process with the new polynomial:
- Divide the leading term [tex]\( 10x^2 \)[/tex] by [tex]\( x \)[/tex]:
[tex]\[ \frac{10x^2}{x} = 10x \][/tex]
- Multiply [tex]\( 10x \)[/tex] by [tex]\( x - 2 \)[/tex]:
[tex]\[ 10x \cdot (x - 2) = 10x^2 - 20x \][/tex]
- Subtract:
[tex]\[ (10x^2 - 5x + 2) - (10x^2 - 20x) = 15x + 2 \][/tex]
5. Repeat the process again:
- Divide the leading term [tex]\( 15x \)[/tex] by [tex]\( x \)[/tex]:
[tex]\[ \frac{15x}{x} = 15 \][/tex]
- Multiply [tex]\( 15 \)[/tex] by [tex]\( x - 2 \)[/tex]:
[tex]\[ 15 \cdot (x - 2) = 15x - 30 \][/tex]
- Subtract:
[tex]\[ (15x + 2) - (15x - 30) = 32 \][/tex]
The quotient is [tex]\( 3x^2 + 10x + 15 \)[/tex] and the remainder is [tex]\( 32 \)[/tex].
### Problem 2:
Dividing [tex]\( 4x^4 - 3x^3 + 2x^2 - x + 6 \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex]
1. Set up the division: To divide [tex]\( 4x^4 - 3x^3 + 2x^2 - x + 6 \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex], we use polynomial long division or synthetic division.
2. Divide the leading term:
- Divide the leading term of the numerator [tex]\( 4x^4 \)[/tex] by the leading term of the denominator [tex]\( 2x^2 \)[/tex]:
[tex]\[ \frac{4x^4}{2x^2} = 2x^2 \][/tex]
3. Multiply and subtract:
- Multiply [tex]\( 2x^2 \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex]:
[tex]\[ 2x^2 \cdot (2x^2 - 1) = 4x^4 - 2x^2 \][/tex]
- Subtract this from the original polynomial:
[tex]\[ (4x^4 - 3x^3 + 2x^2 - x + 6) - (4x^4 - 2x^2) = -3x^3 + 4x^2 - x + 6 \][/tex]
4. Repeat the process with the new polynomial:
- Divide the leading term [tex]\( -3x^3 \)[/tex] by [tex]\( 2x^2 \)[/tex]:
[tex]\[ \frac{-3x^3}{2x^2} = -\frac{3x}{2} \][/tex]
- Multiply [tex]\( -\frac{3x}{2} \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex]:
[tex]\[ -\frac{3x}{2} \cdot (2x^2 - 1) = -3x^3 + \frac{3x}{2} \][/tex]
- Subtract:
[tex]\[ (-3x^3 + 4x^2 - x + 6) - (-3x^3 + \frac{3x}{2}) = 4x^2 - \frac{5x}{2} + 6 \][/tex]
5. Repeat the process again:
- Divide the leading term [tex]\( 4x^2 \)[/tex] by [tex]\( 2x^2 \)[/tex]:
[tex]\[ \frac{4x^2}{2x^2} = 2 \][/tex]
- Multiply [tex]\( 2 \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex]:
[tex]\[ 2 \cdot (2x^2 - 1) = 4x^2 - 2 \][/tex]
- Subtract:
[tex]\[ (4x^2 - \frac{5x}{2} + 6) - (4x^2 - 2) = -\frac{5x}{2} + 8 \][/tex]
The quotient is [tex]\( 2x^2 - \frac{3x}{2} + 2 \)[/tex] and the remainder is [tex]\( 8 - \frac{5x}{2} \)[/tex].
Therefore, the results of the polynomial divisions are:
1. Quotient: [tex]\( 3x^2 + 10x + 15 \)[/tex], Remainder: [tex]\( 32 \)[/tex]
2. Quotient: [tex]\( 2x^2 - \frac{3x}{2} + 2 \)[/tex], Remainder: [tex]\( 8 - \frac{5x}{2} \)[/tex]
### Problem 1:
Dividing [tex]\( 3x^3 + 4x^2 - 5x + 2 \)[/tex] by [tex]\( x - 2 \)[/tex]
1. Set up the division: To divide [tex]\( 3x^3 + 4x^2 - 5x + 2 \)[/tex] by [tex]\( x - 2 \)[/tex], we use polynomial long division.
2. Divide the leading term:
- Divide the leading term of the numerator [tex]\( 3x^3 \)[/tex] by the leading term of the denominator [tex]\( x \)[/tex]:
[tex]\[ \frac{3x^3}{x} = 3x^2 \][/tex]
3. Multiply and subtract:
- Multiply [tex]\( 3x^2 \)[/tex] by [tex]\( x - 2 \)[/tex]:
[tex]\[ 3x^2 \cdot (x - 2) = 3x^3 - 6x^2 \][/tex]
- Subtract this from the original polynomial:
[tex]\[ (3x^3 + 4x^2 - 5x + 2) - (3x^3 - 6x^2) = 10x^2 - 5x + 2 \][/tex]
4. Repeat the process with the new polynomial:
- Divide the leading term [tex]\( 10x^2 \)[/tex] by [tex]\( x \)[/tex]:
[tex]\[ \frac{10x^2}{x} = 10x \][/tex]
- Multiply [tex]\( 10x \)[/tex] by [tex]\( x - 2 \)[/tex]:
[tex]\[ 10x \cdot (x - 2) = 10x^2 - 20x \][/tex]
- Subtract:
[tex]\[ (10x^2 - 5x + 2) - (10x^2 - 20x) = 15x + 2 \][/tex]
5. Repeat the process again:
- Divide the leading term [tex]\( 15x \)[/tex] by [tex]\( x \)[/tex]:
[tex]\[ \frac{15x}{x} = 15 \][/tex]
- Multiply [tex]\( 15 \)[/tex] by [tex]\( x - 2 \)[/tex]:
[tex]\[ 15 \cdot (x - 2) = 15x - 30 \][/tex]
- Subtract:
[tex]\[ (15x + 2) - (15x - 30) = 32 \][/tex]
The quotient is [tex]\( 3x^2 + 10x + 15 \)[/tex] and the remainder is [tex]\( 32 \)[/tex].
### Problem 2:
Dividing [tex]\( 4x^4 - 3x^3 + 2x^2 - x + 6 \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex]
1. Set up the division: To divide [tex]\( 4x^4 - 3x^3 + 2x^2 - x + 6 \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex], we use polynomial long division or synthetic division.
2. Divide the leading term:
- Divide the leading term of the numerator [tex]\( 4x^4 \)[/tex] by the leading term of the denominator [tex]\( 2x^2 \)[/tex]:
[tex]\[ \frac{4x^4}{2x^2} = 2x^2 \][/tex]
3. Multiply and subtract:
- Multiply [tex]\( 2x^2 \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex]:
[tex]\[ 2x^2 \cdot (2x^2 - 1) = 4x^4 - 2x^2 \][/tex]
- Subtract this from the original polynomial:
[tex]\[ (4x^4 - 3x^3 + 2x^2 - x + 6) - (4x^4 - 2x^2) = -3x^3 + 4x^2 - x + 6 \][/tex]
4. Repeat the process with the new polynomial:
- Divide the leading term [tex]\( -3x^3 \)[/tex] by [tex]\( 2x^2 \)[/tex]:
[tex]\[ \frac{-3x^3}{2x^2} = -\frac{3x}{2} \][/tex]
- Multiply [tex]\( -\frac{3x}{2} \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex]:
[tex]\[ -\frac{3x}{2} \cdot (2x^2 - 1) = -3x^3 + \frac{3x}{2} \][/tex]
- Subtract:
[tex]\[ (-3x^3 + 4x^2 - x + 6) - (-3x^3 + \frac{3x}{2}) = 4x^2 - \frac{5x}{2} + 6 \][/tex]
5. Repeat the process again:
- Divide the leading term [tex]\( 4x^2 \)[/tex] by [tex]\( 2x^2 \)[/tex]:
[tex]\[ \frac{4x^2}{2x^2} = 2 \][/tex]
- Multiply [tex]\( 2 \)[/tex] by [tex]\( 2x^2 - 1 \)[/tex]:
[tex]\[ 2 \cdot (2x^2 - 1) = 4x^2 - 2 \][/tex]
- Subtract:
[tex]\[ (4x^2 - \frac{5x}{2} + 6) - (4x^2 - 2) = -\frac{5x}{2} + 8 \][/tex]
The quotient is [tex]\( 2x^2 - \frac{3x}{2} + 2 \)[/tex] and the remainder is [tex]\( 8 - \frac{5x}{2} \)[/tex].
Therefore, the results of the polynomial divisions are:
1. Quotient: [tex]\( 3x^2 + 10x + 15 \)[/tex], Remainder: [tex]\( 32 \)[/tex]
2. Quotient: [tex]\( 2x^2 - \frac{3x}{2} + 2 \)[/tex], Remainder: [tex]\( 8 - \frac{5x}{2} \)[/tex]
