Polynomial Division: Find The Quotient Of (x³ + 3x² + 5x + 3) / (x + 1)

Aug 3, 2025 by Rajiv Sharma 72 views

Decoding Polynomial Division: Finding the Quotient of (x³ + 3x² + 5x + 3) ÷ (x + 1)

Hey there, math enthusiasts! Ever found yourself staring at a polynomial division problem and feeling a little lost? Don't worry, you're not alone! Polynomial division might seem intimidating at first, but with a little guidance, you can conquer it. In this article, we're going to break down one such problem step by step: finding the quotient of $(x^3 + 3x^2 + 5x + 3)

Diving into Polynomial Division

So, what exactly are we trying to achieve here? When we divide one polynomial by another, we're essentially trying to figure out how many times the second polynomial (the divisor) fits into the first polynomial (the dividend). The result of this division is called the quotient, and sometimes we might also have a remainder left over.

Long Division Method

The most common method for polynomial division is the long division method, which is very similar to the long division you learned in elementary school for dividing numbers. Let's walk through the steps involved in solving our problem, $(x^3 + 3x^2 + 5x + 3) ÷ (x + 1)$ .

Step-by-Step Solution

Set up the division:

Write the dividend ( $x^3 + 3x^2 + 5x + 3$ ) inside the division symbol and the divisor ( $x + 1$ ) outside.
```
            _________
x + 1 | x³ + 3x² + 5x + 3
```
Divide the first term:

Divide the first term of the dividend ( $x^3$ ) by the first term of the divisor ( $x$ ). This gives us $x^2$ . Write this above the division symbol, aligned with the $x^2$ term in the dividend.
```
        x²________
x + 1 | x³ + 3x² + 5x + 3
```
Multiply and subtract:

Multiply the quotient term ( $x^2$ ) by the entire divisor ( $x + 1$ ), which gives us $x^3 + x^2$ . Write this below the dividend and subtract it.
```
        x²________
x + 1 | x³ + 3x² + 5x + 3
       -(x³ + x²)
       -----------
            2x² + 5x
```
Bring down the next term:

Bring down the next term from the dividend (+5x) and write it next to the result of the subtraction.
```
        x²________
x + 1 | x³ + 3x² + 5x + 3
       -(x³ + x²)
       -----------
            2x² + 5x
```

Repeat the process:

Divide the first term of the new expression (2x²) by the first term of the divisor (x), which gives us +2x. Write this next to the $x^2$ in the quotient.

        x² + 2x ____
x + 1 | x³ + 3x² + 5x + 3
       -(x³ + x²)
       -----------
            2x² + 5x

Multiply 2x by the divisor (x + 1), which gives us 2x² + 2x. Write this below and subtract.

        x² + 2x ____
x + 1 | x³ + 3x² + 5x + 3
       -(x³ + x²)
       -----------
            2x² + 5x
       -(2x² + 2x)
       -----------
                 3x + 3

Bring down the last term:

Bring down the last term from the dividend (+3).

        x² + 2x ____
x + 1 | x³ + 3x² + 5x + 3
       -(x³ + x²)
       -----------
            2x² + 5x
       -(2x² + 2x)
       -----------
                 3x + 3

Final step:

Divide 3x by x, which gives us +3. Write this in the quotient.

        x² + 2x + 3
x + 1 | x³ + 3x² + 5x + 3
       -(x³ + x²)
       -----------
            2x² + 5x
       -(2x² + 2x)
       -----------
                 3x + 3

Multiply 3 by the divisor (x + 1), which gives us 3x + 3. Subtract this, and we get a remainder of 0.

        x² + 2x + 3
x + 1 | x³ + 3x² + 5x + 3
       -(x³ + x²)
       -----------
            2x² + 5x
       -(2x² + 2x)
       -----------
                 3x + 3
       -(3x + 3)
       -----------
                    0

The Answer

So, the quotient of $(x^3 + 3x^2 + 5x + 3) ÷ (x + 1)$ is $x^2 + 2x + 3$ .

Therefore, the correct answer is C. $x^2 + 2x + 3$ .

Why is Polynomial Division Important?

You might be wondering,