Answered

At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

Perform the division [tex]\((3x^3 - 5x^2 + x - 2) \div (x - 2)\)[/tex].

Select the correct answer.

Sagot :

Sure, let's perform the polynomial division step-by-step to divide [tex]\(3x^3 - 5x^2 + x - 2\)[/tex] by [tex]\(x - 2\)[/tex].

### Step 1: Setup the division
We start by setting up the polynomial division with [tex]\(3x^3 - 5x^2 + x - 2\)[/tex] as the dividend and [tex]\(x - 2\)[/tex] as the divisor.

```
Dividend: 3x^3 - 5x^2 + x - 2
Divisor: x - 2
```

### Step 2: Divide the leading terms
First, divide the leading term of the dividend by the leading term of the divisor:

[tex]\[ \frac{3x^3}{x} = 3x^2 \][/tex]

### Step 3: Multiply and subtract
Next, multiply the entire divisor by [tex]\(3x^2\)[/tex] and subtract this product from the dividend:

1. Multiply:
[tex]\[ (x - 2) \times 3x^2 = 3x^3 - 6x^2 \][/tex]

2. Subtract:
[tex]\[ (3x^3 - 5x^2 + x - 2) - (3x^3 - 6x^2) = (3x^3 - 3x^3) + (-5x^2 + 6x^2) + x - 2 = x^2 + x - 2 \][/tex]

### Step 4: Repeat the process
Continue the same process with the new polynomial [tex]\(x^2 + x - 2\)[/tex]:

1. Divide the leading terms:
[tex]\[ \frac{x^2}{x} = x \][/tex]

2. Multiply:
[tex]\[ (x - 2) \times x = x^2 - 2x \][/tex]

3. Subtract:
[tex]\[ (x^2 + x - 2) - (x^2 - 2x) = (x^2 - x^2) + (x + 2x) - 2 = 3x - 2 \][/tex]

### Step 5: Repeat the process again
Continue with the new polynomial [tex]\(3x - 2\)[/tex]:

1. Divide the leading terms:
[tex]\[ \frac{3x}{x} = 3 \][/tex]

2. Multiply:
[tex]\[ (x - 2) \times 3 = 3x - 6 \][/tex]

3. Subtract:
[tex]\[ (3x - 2) - (3x - 6) = (3x - 3x) + (-2 + 6) = 4 \][/tex]

### Step 6: Compile the quotient and remainder
After completing the division process, we find that:

- Quotient: [tex]\(3x^2 + x + 3\)[/tex]
- Remainder: [tex]\(4\)[/tex]

Therefore, the result of the division [tex]\((3x^3 - 5x^2 + x - 2) \div (x - 2)\)[/tex] is a quotient of [tex]\(3x^2 + x + 3\)[/tex] with a remainder of [tex]\(4\)[/tex].