Answer :
To solve the division of the polynomial [tex]\((x^3 + 2x^2 - 22x - 45)\)[/tex] by [tex]\((x + 5)\)[/tex], we can perform polynomial long division. Let's do it step-by-step:
1. Set up the division: Write [tex]\(x^3 + 2x^2 - 22x - 45\)[/tex] inside the division symbol and [tex]\(x + 5\)[/tex] outside.
2. Divide the first term of the dividend by the first term of the divisor: Divide [tex]\(x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^2\)[/tex]. This quotient is placed above the division bar.
3. Multiply: Multiply [tex]\(x^2\)[/tex] by the entire divisor ([tex]\(x + 5\)[/tex]) to get [tex]\(x^3 + 5x^2\)[/tex].
4. Subtract: Subtract [tex]\(x^3 + 5x^2\)[/tex] from the original polynomial [tex]\((x^3 + 2x^2 - 22x - 45)\)[/tex]. This leaves us with [tex]\(-3x^2 - 22x - 45\)[/tex].
5. Bring down the next term: In this case, we don't need to bring anything down yet as we are left with terms [tex]\(-3x^2\)[/tex].
6. Repeat the process:
- Divide [tex]\(-3x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-3x\)[/tex].
- Multiply [tex]\(-3x\)[/tex] by [tex]\((x + 5)\)[/tex] to get [tex]\(-3x^2 - 15x\)[/tex].
- Subtract again: [tex]\((-3x^2 - 22x) - (-3x^2 - 15x)\)[/tex] gives [tex]\(-7x\)[/tex].
7. Bring down the next term from the original polynomial, which is the [tex]\(-45\)[/tex].
8. Divide [tex]\(-7x - 45\)[/tex]: Divide [tex]\(-7x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-7\)[/tex].
- Multiply [tex]\(-7\)[/tex] by [tex]\((x + 5)\)[/tex] to get [tex]\(-7x - 35\)[/tex].
- Subtract: [tex]\((-7x - 45) - (-7x - 35)\)[/tex] leaves us with [tex]\(-10\)[/tex].
Now, we've completed the division. The quotient is [tex]\(x^2 - 3x - 7\)[/tex], and the remainder is [tex]\(-10\)[/tex].
So, the division can be expressed as:
[tex]\[
x^3 + 2x^2 - 22x - 45 = (x + 5)(x^2 - 3x - 7) - 10
\][/tex]
This means the answer to the division is [tex]\(x^2 - 3x - 7\)[/tex] with a remainder of [tex]\(-10\)[/tex].
1. Set up the division: Write [tex]\(x^3 + 2x^2 - 22x - 45\)[/tex] inside the division symbol and [tex]\(x + 5\)[/tex] outside.
2. Divide the first term of the dividend by the first term of the divisor: Divide [tex]\(x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^2\)[/tex]. This quotient is placed above the division bar.
3. Multiply: Multiply [tex]\(x^2\)[/tex] by the entire divisor ([tex]\(x + 5\)[/tex]) to get [tex]\(x^3 + 5x^2\)[/tex].
4. Subtract: Subtract [tex]\(x^3 + 5x^2\)[/tex] from the original polynomial [tex]\((x^3 + 2x^2 - 22x - 45)\)[/tex]. This leaves us with [tex]\(-3x^2 - 22x - 45\)[/tex].
5. Bring down the next term: In this case, we don't need to bring anything down yet as we are left with terms [tex]\(-3x^2\)[/tex].
6. Repeat the process:
- Divide [tex]\(-3x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-3x\)[/tex].
- Multiply [tex]\(-3x\)[/tex] by [tex]\((x + 5)\)[/tex] to get [tex]\(-3x^2 - 15x\)[/tex].
- Subtract again: [tex]\((-3x^2 - 22x) - (-3x^2 - 15x)\)[/tex] gives [tex]\(-7x\)[/tex].
7. Bring down the next term from the original polynomial, which is the [tex]\(-45\)[/tex].
8. Divide [tex]\(-7x - 45\)[/tex]: Divide [tex]\(-7x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-7\)[/tex].
- Multiply [tex]\(-7\)[/tex] by [tex]\((x + 5)\)[/tex] to get [tex]\(-7x - 35\)[/tex].
- Subtract: [tex]\((-7x - 45) - (-7x - 35)\)[/tex] leaves us with [tex]\(-10\)[/tex].
Now, we've completed the division. The quotient is [tex]\(x^2 - 3x - 7\)[/tex], and the remainder is [tex]\(-10\)[/tex].
So, the division can be expressed as:
[tex]\[
x^3 + 2x^2 - 22x - 45 = (x + 5)(x^2 - 3x - 7) - 10
\][/tex]
This means the answer to the division is [tex]\(x^2 - 3x - 7\)[/tex] with a remainder of [tex]\(-10\)[/tex].
