Answer :
To divide the polynomial [tex]\(35x^9 - 42x^8 - 5x^5 + 6x^4\)[/tex] by [tex]\(5x^2 - 6x\)[/tex] using long division, we'll perform the division step-by-step:
1. Set Up the Division: We're dividing [tex]\(35x^9 - 42x^8 - 5x^5 + 6x^4\)[/tex] by [tex]\(5x^2 - 6x\)[/tex].
[tex]\[ \frac{35x^9 - 42x^8 - 5x^5 + 6x^4}{5x^2 - 6x} \][/tex]
2. Divide the Leading Terms:
- Divide the leading term of the numerator [tex]\(35x^9\)[/tex] by the leading term of the denominator [tex]\(5x^2\)[/tex]:
[tex]\[ \frac{35x^9}{5x^2} = 7x^7 \][/tex]
3. Multiply and Subtract:
- Multiply [tex]\(7x^7\)[/tex] by the entire divisor [tex]\(5x^2 - 6x\)[/tex]:
[tex]\[ 7x^7 \cdot (5x^2 - 6x) = 35x^9 - 42x^8 \][/tex]
- Subtract this result from the original polynomial:
[tex]\[ (35x^9 - 42x^8 - 5x^5 + 6x^4) - (35x^9 - 42x^8) = -5x^5 + 6x^4 \][/tex]
4. Repeat the Process:
- Now we need to bring down terms if necessary and repeat the process. Since the next term to consider is [tex]\(-5x^5\)[/tex]:
[tex]\[ \frac{(-5x^5)}{5x^2} = -x^3 \][/tex]
- Multiply [tex]\(-x^3\)[/tex] by [tex]\(5x^2 - 6x\)[/tex]:
[tex]\[ -x^3 \cdot (5x^2 - 6x) = -5x^5 + 6x^4 \][/tex]
- Subtract this result:
[tex]\[ (-5x^5 + 6x^4) - (-5x^5 + 6x^4) = 0 \][/tex]
Since there are no remaining terms of lower degree of [tex]\(x\)[/tex] that [tex]\(5x^2 - 6x\)[/tex] can divide into, we can conclude the process.
5. Combine the Results:
- Summarize the polynomial quotient and remainder:
The quotient is:
[tex]\[ 7x^7 - x^3 \][/tex]
There is no remainder (remainder is 0).
6. Write the Answer in Standard Form:
The quotient polynomial in standard form is:
[tex]\[ \boxed{7x^7 - x^3} \][/tex]
And the remainder is:
[tex]\[ \boxed{0} \][/tex]
Thus, the final answer is:
[tex]\[ 7x^7 - x^3 \text{ with a remainder of } 0. \][/tex]
1. Set Up the Division: We're dividing [tex]\(35x^9 - 42x^8 - 5x^5 + 6x^4\)[/tex] by [tex]\(5x^2 - 6x\)[/tex].
[tex]\[ \frac{35x^9 - 42x^8 - 5x^5 + 6x^4}{5x^2 - 6x} \][/tex]
2. Divide the Leading Terms:
- Divide the leading term of the numerator [tex]\(35x^9\)[/tex] by the leading term of the denominator [tex]\(5x^2\)[/tex]:
[tex]\[ \frac{35x^9}{5x^2} = 7x^7 \][/tex]
3. Multiply and Subtract:
- Multiply [tex]\(7x^7\)[/tex] by the entire divisor [tex]\(5x^2 - 6x\)[/tex]:
[tex]\[ 7x^7 \cdot (5x^2 - 6x) = 35x^9 - 42x^8 \][/tex]
- Subtract this result from the original polynomial:
[tex]\[ (35x^9 - 42x^8 - 5x^5 + 6x^4) - (35x^9 - 42x^8) = -5x^5 + 6x^4 \][/tex]
4. Repeat the Process:
- Now we need to bring down terms if necessary and repeat the process. Since the next term to consider is [tex]\(-5x^5\)[/tex]:
[tex]\[ \frac{(-5x^5)}{5x^2} = -x^3 \][/tex]
- Multiply [tex]\(-x^3\)[/tex] by [tex]\(5x^2 - 6x\)[/tex]:
[tex]\[ -x^3 \cdot (5x^2 - 6x) = -5x^5 + 6x^4 \][/tex]
- Subtract this result:
[tex]\[ (-5x^5 + 6x^4) - (-5x^5 + 6x^4) = 0 \][/tex]
Since there are no remaining terms of lower degree of [tex]\(x\)[/tex] that [tex]\(5x^2 - 6x\)[/tex] can divide into, we can conclude the process.
5. Combine the Results:
- Summarize the polynomial quotient and remainder:
The quotient is:
[tex]\[ 7x^7 - x^3 \][/tex]
There is no remainder (remainder is 0).
6. Write the Answer in Standard Form:
The quotient polynomial in standard form is:
[tex]\[ \boxed{7x^7 - x^3} \][/tex]
And the remainder is:
[tex]\[ \boxed{0} \][/tex]
Thus, the final answer is:
[tex]\[ 7x^7 - x^3 \text{ with a remainder of } 0. \][/tex]
