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Answer :
To solve the problem of dividing the polynomial [tex]\(3x^3 + 19x^2 - 21x - 4\)[/tex] by [tex]\(3x + 1\)[/tex], you will perform polynomial division, similar to long division with numbers. Here is a detailed, step-by-step solution:
1. Set up the division: Write the dividend (the polynomial you're dividing) under the division symbol, and place the divisor [tex]\(3x + 1\)[/tex] outside.
2. Divide the first terms: Look at the leading term of the dividend, which is [tex]\(3x^3\)[/tex]. Divide it by the leading term of the divisor, [tex]\(3x\)[/tex]. This gives you [tex]\(x^2\)[/tex]. Write [tex]\(x^2\)[/tex] above the division bar as the first term of the quotient.
3. Multiply and subtract: Multiply the entire divisor [tex]\(3x + 1\)[/tex] by the [tex]\(x^2\)[/tex] you just found and subtract the result from the original polynomial:
[tex]\[
(3x + 1) \times x^2 = 3x^3 + x^2
\][/tex]
Subtract:
[tex]\[
(3x^3 + 19x^2 - 21x - 4) - (3x^3 + x^2) = 18x^2 - 21x - 4
\][/tex]
4. Bring down the next term: Now, bring down the next term, which keeps the polynomial [tex]\(18x^2 - 21x - 4\)[/tex].
5. Repeat the process: Divide the leading term of this new polynomial [tex]\(18x^2\)[/tex] by [tex]\(3x\)[/tex] (the leading term of the divisor), which gives [tex]\(6x\)[/tex]. Write this in the quotient next to [tex]\(x^2\)[/tex].
6. Multiply and subtract: Multiply the entire divisor [tex]\(3x + 1\)[/tex] by [tex]\(6x\)[/tex] and subtract:
[tex]\[
(3x + 1) \times 6x = 18x^2 + 6x
\][/tex]
Subtract:
[tex]\[
(18x^2 - 21x - 4) - (18x^2 + 6x) = -27x - 4
\][/tex]
7. Bring down the next term: Again, bring down the next term, giving you [tex]\(-27x - 4\)[/tex].
8. One more round: Divide the leading term [tex]\(-27x\)[/tex] by [tex]\(3x\)[/tex], which gives [tex]\(-9\)[/tex]. Write this in the quotient.
9. Final multiply and subtract: Multiply the entire divisor:
[tex]\[
(3x + 1) \times -9 = -27x - 9
\][/tex]
Subtract:
[tex]\[
(-27x - 4) - (-27x - 9) = 5
\][/tex]
10. Conclusion: With no more terms to bring down, [tex]\(5\)[/tex] is the remainder.
Thus, the quotient of the division is [tex]\(x^2 + 6x - 9\)[/tex] and the remainder is [tex]\(5\)[/tex].
In conclusion, the result of the polynomial division is:
[tex]\[
\frac{3x^3 + 19x^2 - 21x - 4}{3x + 1} = x^2 + 6x - 9 \quad \text{with a remainder of } 5
\][/tex]
1. Set up the division: Write the dividend (the polynomial you're dividing) under the division symbol, and place the divisor [tex]\(3x + 1\)[/tex] outside.
2. Divide the first terms: Look at the leading term of the dividend, which is [tex]\(3x^3\)[/tex]. Divide it by the leading term of the divisor, [tex]\(3x\)[/tex]. This gives you [tex]\(x^2\)[/tex]. Write [tex]\(x^2\)[/tex] above the division bar as the first term of the quotient.
3. Multiply and subtract: Multiply the entire divisor [tex]\(3x + 1\)[/tex] by the [tex]\(x^2\)[/tex] you just found and subtract the result from the original polynomial:
[tex]\[
(3x + 1) \times x^2 = 3x^3 + x^2
\][/tex]
Subtract:
[tex]\[
(3x^3 + 19x^2 - 21x - 4) - (3x^3 + x^2) = 18x^2 - 21x - 4
\][/tex]
4. Bring down the next term: Now, bring down the next term, which keeps the polynomial [tex]\(18x^2 - 21x - 4\)[/tex].
5. Repeat the process: Divide the leading term of this new polynomial [tex]\(18x^2\)[/tex] by [tex]\(3x\)[/tex] (the leading term of the divisor), which gives [tex]\(6x\)[/tex]. Write this in the quotient next to [tex]\(x^2\)[/tex].
6. Multiply and subtract: Multiply the entire divisor [tex]\(3x + 1\)[/tex] by [tex]\(6x\)[/tex] and subtract:
[tex]\[
(3x + 1) \times 6x = 18x^2 + 6x
\][/tex]
Subtract:
[tex]\[
(18x^2 - 21x - 4) - (18x^2 + 6x) = -27x - 4
\][/tex]
7. Bring down the next term: Again, bring down the next term, giving you [tex]\(-27x - 4\)[/tex].
8. One more round: Divide the leading term [tex]\(-27x\)[/tex] by [tex]\(3x\)[/tex], which gives [tex]\(-9\)[/tex]. Write this in the quotient.
9. Final multiply and subtract: Multiply the entire divisor:
[tex]\[
(3x + 1) \times -9 = -27x - 9
\][/tex]
Subtract:
[tex]\[
(-27x - 4) - (-27x - 9) = 5
\][/tex]
10. Conclusion: With no more terms to bring down, [tex]\(5\)[/tex] is the remainder.
Thus, the quotient of the division is [tex]\(x^2 + 6x - 9\)[/tex] and the remainder is [tex]\(5\)[/tex].
In conclusion, the result of the polynomial division is:
[tex]\[
\frac{3x^3 + 19x^2 - 21x - 4}{3x + 1} = x^2 + 6x - 9 \quad \text{with a remainder of } 5
\][/tex]
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