Answer :
To find the quotient and remainder for the division of the polynomial [tex]\(5x^4 - 3x^2 - 4x + 6\)[/tex] by [tex]\(x - 7\)[/tex], we can use polynomial long division. Here’s a step-by-step process:
1. Set up the division: Write [tex]\(5x^4 - 3x^2 - 4x + 6\)[/tex] under a long division bracket, and [tex]\(x - 7\)[/tex] outside.
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(5x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives you [tex]\(5x^3\)[/tex]. Write this as the first term of the quotient above the division bracket.
3. Multiply and subtract: Multiply the entire divisor [tex]\(x - 7\)[/tex] by the term you just found ([tex]\(5x^3\)[/tex]), which gives [tex]\(5x^4 - 35x^3\)[/tex]. Subtract this from the original dividend to find the new dividend:
[tex]\[
(5x^4 - 3x^2 - 4x + 6) - (5x^4 - 35x^3) = 35x^3 - 3x^2 - 4x + 6.
\][/tex]
4. Repeat the process:
- Divide the leading term of the new dividend ([tex]\(35x^3\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]), resulting in [tex]\(35x^2\)[/tex]. Add this to the quotient.
- Multiply and subtract again: Multiply [tex]\(x - 7\)[/tex] by [tex]\(35x^2\)[/tex] to get [tex]\(35x^3 - 245x^2\)[/tex], and subtract:
[tex]\[
(35x^3 - 3x^2 - 4x + 6) - (35x^3 - 245x^2) = 242x^2 - 4x + 6.
\][/tex]
5. Continue dividing:
- Divide [tex]\(242x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(242x\)[/tex], and add to the quotient.
- Multiply and subtract: [tex]\(242x \cdot (x - 7) = 242x^2 - 1694x\)[/tex], then subtract:
[tex]\[
(242x^2 - 4x + 6) - (242x^2 - 1694x) = 1690x + 6.
\][/tex]
6. Final division:
- Divide [tex]\(1690x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(1690\)[/tex]. Add to the quotient.
- Multiply and subtract: [tex]\(1690 \cdot (x - 7) = 1690x - 11830\)[/tex], then subtract:
[tex]\[
(1690x + 6) - (1690x - 11830) = 11836.
\][/tex]
7. Conclusion: The quotient of the division is [tex]\(5x^3 + 35x^2 + 242x + 1690\)[/tex], and the remainder is [tex]\(11836\)[/tex].
So, the answer is [tex]\((\text{Quotient: } 5x^3 + 35x^2 + 242x + 1690, \text{ Remainder: } 11836)\)[/tex]. Therefore, the correct choice is:
[tex]\( \boxed{d. \ 5x^3 + 35x^2 + 242x + 1690; \ 11836} \)[/tex]
1. Set up the division: Write [tex]\(5x^4 - 3x^2 - 4x + 6\)[/tex] under a long division bracket, and [tex]\(x - 7\)[/tex] outside.
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(5x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives you [tex]\(5x^3\)[/tex]. Write this as the first term of the quotient above the division bracket.
3. Multiply and subtract: Multiply the entire divisor [tex]\(x - 7\)[/tex] by the term you just found ([tex]\(5x^3\)[/tex]), which gives [tex]\(5x^4 - 35x^3\)[/tex]. Subtract this from the original dividend to find the new dividend:
[tex]\[
(5x^4 - 3x^2 - 4x + 6) - (5x^4 - 35x^3) = 35x^3 - 3x^2 - 4x + 6.
\][/tex]
4. Repeat the process:
- Divide the leading term of the new dividend ([tex]\(35x^3\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]), resulting in [tex]\(35x^2\)[/tex]. Add this to the quotient.
- Multiply and subtract again: Multiply [tex]\(x - 7\)[/tex] by [tex]\(35x^2\)[/tex] to get [tex]\(35x^3 - 245x^2\)[/tex], and subtract:
[tex]\[
(35x^3 - 3x^2 - 4x + 6) - (35x^3 - 245x^2) = 242x^2 - 4x + 6.
\][/tex]
5. Continue dividing:
- Divide [tex]\(242x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(242x\)[/tex], and add to the quotient.
- Multiply and subtract: [tex]\(242x \cdot (x - 7) = 242x^2 - 1694x\)[/tex], then subtract:
[tex]\[
(242x^2 - 4x + 6) - (242x^2 - 1694x) = 1690x + 6.
\][/tex]
6. Final division:
- Divide [tex]\(1690x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(1690\)[/tex]. Add to the quotient.
- Multiply and subtract: [tex]\(1690 \cdot (x - 7) = 1690x - 11830\)[/tex], then subtract:
[tex]\[
(1690x + 6) - (1690x - 11830) = 11836.
\][/tex]
7. Conclusion: The quotient of the division is [tex]\(5x^3 + 35x^2 + 242x + 1690\)[/tex], and the remainder is [tex]\(11836\)[/tex].
So, the answer is [tex]\((\text{Quotient: } 5x^3 + 35x^2 + 242x + 1690, \text{ Remainder: } 11836)\)[/tex]. Therefore, the correct choice is:
[tex]\( \boxed{d. \ 5x^3 + 35x^2 + 242x + 1690; \ 11836} \)[/tex]
