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Answer :
To divide the polynomial [tex]\(35x^9 - 42x^8 - 5x^5 + 6x^4\)[/tex] by [tex]\(5x^2 - 6x\)[/tex] using polynomial long division, follow these steps:
### Step-by-Step Solution
1. Set up the division:
[tex]\[
\text{Dividend: } 35x^9 - 42x^8 - 5x^5 + 6x^4
\][/tex]
[tex]\[
\text{Divisor: } 5x^2 - 6x
\][/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{35x^9}{5x^2} = 7x^7
\][/tex]
This is the first term of the quotient.
3. Multiply the entire divisor by the first term of the quotient:
[tex]\[
(5x^2 - 6x) \cdot 7x^7 = 35x^9 - 42x^8
\][/tex]
4. Subtract this product from the original dividend:
[tex]\[
(35x^9 - 42x^8 - 5x^5 + 6x^4) - (35x^9 - 42x^8) = -5x^5 + 6x^4
\][/tex]
The new dividend is now: [tex]\( -5x^5 + 6x^4 \)[/tex]
5. Repeat the division process with the new dividend:
[tex]\[
\frac{-5x^5}{5x^2} = -x^3
\][/tex]
This is the next term of the quotient.
6. Multiply the entire divisor by this new term of the quotient:
[tex]\[
(5x^2 - 6x) \cdot -x^3 = -5x^5 + 6x^4
\][/tex]
7. Subtract this product from the current dividend:
[tex]\[
(-5x^5 + 6x^4) - (-5x^5 + 6x^4) = 0
\][/tex]
At this point, the new dividend is zero, and we've finished the division process. The quotient is:
[tex]\[
7x^7 - x^3
\][/tex]
And the remainder is:
[tex]\[
0
\][/tex]
### Final Answer
The quotient is:
[tex]\[
7x^7 - x^3
\][/tex]
And the remainder is:
[tex]\[
0
\][/tex]
So, when the polynomial [tex]\(35x^9 - 42x^8 - 5x^5 + 6x^4\)[/tex] is divided by [tex]\(5x^2 - 6x\)[/tex], the result is:
[tex]\[
\boxed{(7x^7 - x^3, 0)}
\][/tex]
### Step-by-Step Solution
1. Set up the division:
[tex]\[
\text{Dividend: } 35x^9 - 42x^8 - 5x^5 + 6x^4
\][/tex]
[tex]\[
\text{Divisor: } 5x^2 - 6x
\][/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{35x^9}{5x^2} = 7x^7
\][/tex]
This is the first term of the quotient.
3. Multiply the entire divisor by the first term of the quotient:
[tex]\[
(5x^2 - 6x) \cdot 7x^7 = 35x^9 - 42x^8
\][/tex]
4. Subtract this product from the original dividend:
[tex]\[
(35x^9 - 42x^8 - 5x^5 + 6x^4) - (35x^9 - 42x^8) = -5x^5 + 6x^4
\][/tex]
The new dividend is now: [tex]\( -5x^5 + 6x^4 \)[/tex]
5. Repeat the division process with the new dividend:
[tex]\[
\frac{-5x^5}{5x^2} = -x^3
\][/tex]
This is the next term of the quotient.
6. Multiply the entire divisor by this new term of the quotient:
[tex]\[
(5x^2 - 6x) \cdot -x^3 = -5x^5 + 6x^4
\][/tex]
7. Subtract this product from the current dividend:
[tex]\[
(-5x^5 + 6x^4) - (-5x^5 + 6x^4) = 0
\][/tex]
At this point, the new dividend is zero, and we've finished the division process. The quotient is:
[tex]\[
7x^7 - x^3
\][/tex]
And the remainder is:
[tex]\[
0
\][/tex]
### Final Answer
The quotient is:
[tex]\[
7x^7 - x^3
\][/tex]
And the remainder is:
[tex]\[
0
\][/tex]
So, when the polynomial [tex]\(35x^9 - 42x^8 - 5x^5 + 6x^4\)[/tex] is divided by [tex]\(5x^2 - 6x\)[/tex], the result is:
[tex]\[
\boxed{(7x^7 - x^3, 0)}
\][/tex]
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