Answer :
To divide the polynomial [tex]\(x^3 + 16x^2 + 64x + 4\)[/tex] by [tex]\(x + 7\)[/tex], we will use polynomial long division. Here are the steps:
1. Set up the division: Write the dividend [tex]\(x^3 + 16x^2 + 64x + 4\)[/tex] under the long division symbol and the divisor [tex]\(x + 7\)[/tex] outside.
2. Divide the first term: Divide the first term of the dividend [tex]\(x^3\)[/tex] by the first term of the divisor [tex]\(x\)[/tex], which gives [tex]\(x^2\)[/tex]. This is the first term of the quotient.
3. Multiply and subtract: Multiply [tex]\(x^2\)[/tex] by the entire divisor [tex]\(x + 7\)[/tex] to get [tex]\(x^3 + 7x^2\)[/tex]. Subtract this from the original dividend to get the new dividend:
[tex]\((x^3 + 16x^2 + 64x + 4) - (x^3 + 7x^2) = 9x^2 + 64x + 4\)[/tex].
4. Repeat the process:
- Divide the first term of the new dividend [tex]\(9x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(9x\)[/tex].
- Multiply [tex]\(9x\)[/tex] by the divisor [tex]\(x + 7\)[/tex] to obtain [tex]\(9x^2 + 63x\)[/tex].
- Subtract from the new dividend:
[tex]\((9x^2 + 64x + 4) - (9x^2 + 63x) = x + 4\)[/tex].
5. Continue the process:
- Divide [tex]\(x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(1\)[/tex].
- Multiply [tex]\(1\)[/tex] by [tex]\(x + 7\)[/tex] to get [tex]\(x + 7\)[/tex].
- Subtract from the current dividend:
[tex]\((x + 4) - (x + 7) = -3\)[/tex].
6. Write the result: The quotient is [tex]\(x^2 + 9x + 1\)[/tex] and the remainder is [tex]\(-3\)[/tex].
So, the division of the polynomial [tex]\(x^3 + 16x^2 + 64x + 4\)[/tex] by [tex]\(x + 7\)[/tex] results in a quotient of [tex]\(x^2 + 9x + 1\)[/tex] with a remainder of [tex]\(-3\)[/tex].
1. Set up the division: Write the dividend [tex]\(x^3 + 16x^2 + 64x + 4\)[/tex] under the long division symbol and the divisor [tex]\(x + 7\)[/tex] outside.
2. Divide the first term: Divide the first term of the dividend [tex]\(x^3\)[/tex] by the first term of the divisor [tex]\(x\)[/tex], which gives [tex]\(x^2\)[/tex]. This is the first term of the quotient.
3. Multiply and subtract: Multiply [tex]\(x^2\)[/tex] by the entire divisor [tex]\(x + 7\)[/tex] to get [tex]\(x^3 + 7x^2\)[/tex]. Subtract this from the original dividend to get the new dividend:
[tex]\((x^3 + 16x^2 + 64x + 4) - (x^3 + 7x^2) = 9x^2 + 64x + 4\)[/tex].
4. Repeat the process:
- Divide the first term of the new dividend [tex]\(9x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(9x\)[/tex].
- Multiply [tex]\(9x\)[/tex] by the divisor [tex]\(x + 7\)[/tex] to obtain [tex]\(9x^2 + 63x\)[/tex].
- Subtract from the new dividend:
[tex]\((9x^2 + 64x + 4) - (9x^2 + 63x) = x + 4\)[/tex].
5. Continue the process:
- Divide [tex]\(x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(1\)[/tex].
- Multiply [tex]\(1\)[/tex] by [tex]\(x + 7\)[/tex] to get [tex]\(x + 7\)[/tex].
- Subtract from the current dividend:
[tex]\((x + 4) - (x + 7) = -3\)[/tex].
6. Write the result: The quotient is [tex]\(x^2 + 9x + 1\)[/tex] and the remainder is [tex]\(-3\)[/tex].
So, the division of the polynomial [tex]\(x^3 + 16x^2 + 64x + 4\)[/tex] by [tex]\(x + 7\)[/tex] results in a quotient of [tex]\(x^2 + 9x + 1\)[/tex] with a remainder of [tex]\(-3\)[/tex].
