Answer :
To perform the division [tex]\((4x^4 + 19x^3 + 24x^2 + 9x - 7) \div (x + 3)\)[/tex] and write the answer in fraction form, we use polynomial long division. Here are the steps to get to the solution:
1. Set Up the Division:
- Write the dividend [tex]\( 4x^4 + 19x^3 + 24x^2 + 9x - 7 \)[/tex].
- Write the divisor [tex]\( x + 3 \)[/tex].
2. Perform Polynomial Long Division:
- Divide the first terms: Divide [tex]\( 4x^4 \)[/tex] by [tex]\( x \)[/tex]. The result is [tex]\( 4x^3 \)[/tex].
- Multiply: Multiply [tex]\( 4x^3 \)[/tex] by [tex]\( x + 3 \)[/tex] to get [tex]\( 4x^4 + 12x^3 \)[/tex].
- Subtract: Subtract [tex]\( 4x^4 + 12x^3 \)[/tex] from the dividend, which yields [tex]\( 7x^3 + 24x^2 + 9x - 7 \)[/tex].
- Repeat the process with the new polynomial [tex]\( 7x^3 + 24x^2 + 9x - 7 \)[/tex]:
- Divide [tex]\( 7x^3 \)[/tex] by [tex]\( x \)[/tex]. The result is [tex]\( 7x^2 \)[/tex].
- Multiply [tex]\( 7x^2 \)[/tex] by [tex]\( x + 3 \)[/tex] to get [tex]\( 7x^3 + 21x^2 \)[/tex].
- Subtract [tex]\( 7x^3 + 21x^2 \)[/tex] to obtain [tex]\( 3x^2 + 9x - 7 \)[/tex].
- Continue with [tex]\( 3x^2 + 9x - 7 \)[/tex]:
- Divide [tex]\( 3x^2 \)[/tex] by [tex]\( x \)[/tex]. The result is [tex]\( 3x \)[/tex].
- Multiply [tex]\( 3x \)[/tex] by [tex]\( x + 3 \)[/tex] to get [tex]\( 3x^2 + 9x \)[/tex].
- Subtract [tex]\( 3x^2 + 9x \)[/tex] to find [tex]\( -7 \)[/tex].
- Handle the remainder:
- Divide [tex]\( -7 \)[/tex] by [tex]\( x + 3 \)[/tex]. Since there are no further variables, [tex]\(-7\)[/tex] is the remainder.
3. Combine Results:
The quotient from the division is given by the terms obtained: [tex]\( 4x^3 + 7x^2 + 3x \)[/tex].
4. Express the Final Answer:
- The quotient is [tex]\( 4x^3 + 7x^2 + 3x \)[/tex] with a remainder of [tex]\(-7\)[/tex].
- Therefore, the division can be expressed as [tex]\( \frac{4x^4 + 19x^3 + 24x^2 + 9x - 7}{x + 3} = 4x^3 + 7x^2 + 3x + \frac{-7}{x+3} \)[/tex].
Thus, when dividing the polynomial by [tex]\(x + 3\)[/tex], the answer is [tex]\(4x^3 + 7x^2 + 3x - \frac{7}{x+3}\)[/tex].
1. Set Up the Division:
- Write the dividend [tex]\( 4x^4 + 19x^3 + 24x^2 + 9x - 7 \)[/tex].
- Write the divisor [tex]\( x + 3 \)[/tex].
2. Perform Polynomial Long Division:
- Divide the first terms: Divide [tex]\( 4x^4 \)[/tex] by [tex]\( x \)[/tex]. The result is [tex]\( 4x^3 \)[/tex].
- Multiply: Multiply [tex]\( 4x^3 \)[/tex] by [tex]\( x + 3 \)[/tex] to get [tex]\( 4x^4 + 12x^3 \)[/tex].
- Subtract: Subtract [tex]\( 4x^4 + 12x^3 \)[/tex] from the dividend, which yields [tex]\( 7x^3 + 24x^2 + 9x - 7 \)[/tex].
- Repeat the process with the new polynomial [tex]\( 7x^3 + 24x^2 + 9x - 7 \)[/tex]:
- Divide [tex]\( 7x^3 \)[/tex] by [tex]\( x \)[/tex]. The result is [tex]\( 7x^2 \)[/tex].
- Multiply [tex]\( 7x^2 \)[/tex] by [tex]\( x + 3 \)[/tex] to get [tex]\( 7x^3 + 21x^2 \)[/tex].
- Subtract [tex]\( 7x^3 + 21x^2 \)[/tex] to obtain [tex]\( 3x^2 + 9x - 7 \)[/tex].
- Continue with [tex]\( 3x^2 + 9x - 7 \)[/tex]:
- Divide [tex]\( 3x^2 \)[/tex] by [tex]\( x \)[/tex]. The result is [tex]\( 3x \)[/tex].
- Multiply [tex]\( 3x \)[/tex] by [tex]\( x + 3 \)[/tex] to get [tex]\( 3x^2 + 9x \)[/tex].
- Subtract [tex]\( 3x^2 + 9x \)[/tex] to find [tex]\( -7 \)[/tex].
- Handle the remainder:
- Divide [tex]\( -7 \)[/tex] by [tex]\( x + 3 \)[/tex]. Since there are no further variables, [tex]\(-7\)[/tex] is the remainder.
3. Combine Results:
The quotient from the division is given by the terms obtained: [tex]\( 4x^3 + 7x^2 + 3x \)[/tex].
4. Express the Final Answer:
- The quotient is [tex]\( 4x^3 + 7x^2 + 3x \)[/tex] with a remainder of [tex]\(-7\)[/tex].
- Therefore, the division can be expressed as [tex]\( \frac{4x^4 + 19x^3 + 24x^2 + 9x - 7}{x + 3} = 4x^3 + 7x^2 + 3x + \frac{-7}{x+3} \)[/tex].
Thus, when dividing the polynomial by [tex]\(x + 3\)[/tex], the answer is [tex]\(4x^3 + 7x^2 + 3x - \frac{7}{x+3}\)[/tex].
