Answer :
To solve the problem of dividing the polynomial [tex]\(18x^4 - 9x^3 + 21x^2\)[/tex] by [tex]\(3x\)[/tex], we need to perform polynomial division. Here’s a step-by-step explanation:
1. Set up the Division:
We have the polynomial [tex]\(18x^4 - 9x^3 + 21x^2\)[/tex] which we want to divide by [tex]\(3x\)[/tex].
2. Divide Each Term:
- Start with the first term: [tex]\(\frac{18x^4}{3x}\)[/tex].
- Divide the coefficients: [tex]\( \frac{18}{3} = 6 \)[/tex].
- Subtract the exponents of [tex]\(x\)[/tex]: [tex]\( 4 - 1 = 3 \)[/tex].
- Result: [tex]\(6x^3\)[/tex].
- Move to the second term: [tex]\(\frac{-9x^3}{3x}\)[/tex].
- Divide the coefficients: [tex]\(-9 \div 3 = -3\)[/tex].
- Subtract the exponents: [tex]\(3 - 1 = 2\)[/tex].
- Result: [tex]\(-3x^2\)[/tex].
- Finally, the third term: [tex]\(\frac{21x^2}{3x}\)[/tex].
- Divide the coefficients: [tex]\(21 \div 3 = 7\)[/tex].
- Subtract the exponents: [tex]\(2 - 1 = 1\)[/tex].
- Result: [tex]\(7x\)[/tex].
3. Construct the Quotient:
- Combine the results from each division step:
- The quotient is [tex]\(6x^3 - 3x^2 + 7x\)[/tex].
4. Check the Remainder:
- In this division, the remainder is [tex]\(0\)[/tex] because each term in the polynomial [tex]\(18x^4 - 9x^3 + 21x^2\)[/tex] divides evenly by [tex]\(3x\)[/tex].
5. Final Result:
- The quotient of the division is [tex]\(6x^3 - 3x^2 + 7x\)[/tex].
- The remainder is [tex]\(0\)[/tex].
Therefore, the result of dividing [tex]\(18x^4 - 9x^3 + 21x^2\)[/tex] by [tex]\(3x\)[/tex] is [tex]\(6x^3 - 3x^2 + 7x\)[/tex] with no remainder.
1. Set up the Division:
We have the polynomial [tex]\(18x^4 - 9x^3 + 21x^2\)[/tex] which we want to divide by [tex]\(3x\)[/tex].
2. Divide Each Term:
- Start with the first term: [tex]\(\frac{18x^4}{3x}\)[/tex].
- Divide the coefficients: [tex]\( \frac{18}{3} = 6 \)[/tex].
- Subtract the exponents of [tex]\(x\)[/tex]: [tex]\( 4 - 1 = 3 \)[/tex].
- Result: [tex]\(6x^3\)[/tex].
- Move to the second term: [tex]\(\frac{-9x^3}{3x}\)[/tex].
- Divide the coefficients: [tex]\(-9 \div 3 = -3\)[/tex].
- Subtract the exponents: [tex]\(3 - 1 = 2\)[/tex].
- Result: [tex]\(-3x^2\)[/tex].
- Finally, the third term: [tex]\(\frac{21x^2}{3x}\)[/tex].
- Divide the coefficients: [tex]\(21 \div 3 = 7\)[/tex].
- Subtract the exponents: [tex]\(2 - 1 = 1\)[/tex].
- Result: [tex]\(7x\)[/tex].
3. Construct the Quotient:
- Combine the results from each division step:
- The quotient is [tex]\(6x^3 - 3x^2 + 7x\)[/tex].
4. Check the Remainder:
- In this division, the remainder is [tex]\(0\)[/tex] because each term in the polynomial [tex]\(18x^4 - 9x^3 + 21x^2\)[/tex] divides evenly by [tex]\(3x\)[/tex].
5. Final Result:
- The quotient of the division is [tex]\(6x^3 - 3x^2 + 7x\)[/tex].
- The remainder is [tex]\(0\)[/tex].
Therefore, the result of dividing [tex]\(18x^4 - 9x^3 + 21x^2\)[/tex] by [tex]\(3x\)[/tex] is [tex]\(6x^3 - 3x^2 + 7x\)[/tex] with no remainder.
