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Answer :
Sure! Let's go through a step-by-step solution to divide the polynomial [tex]\((2x^4 - 2x^3 - 19x^2 + 11x + 3)\)[/tex] by the polynomial [tex]\((2x^2 - 2x - 1)\)[/tex].
### Step 1: Set Up the Division
Write the dividend ([tex]\(2x^4 - 2x^3 - 19x^2 + 11x + 3\)[/tex]) and the divisor ([tex]\(2x^2 - 2x - 1\)[/tex]) for polynomial long division.
### Step 2: Divide the Leading Terms
- Look at the leading term of the dividend [tex]\(2x^4\)[/tex] and the leading term of the divisor [tex]\(2x^2\)[/tex].
- Divide [tex]\(2x^4\)[/tex] by [tex]\(2x^2\)[/tex] to get the first term of the quotient, [tex]\(x^2\)[/tex].
### Step 3: Multiply and Subtract
- Multiply the entire divisor [tex]\(2x^2 - 2x - 1\)[/tex] by the term just found, [tex]\(x^2\)[/tex], to get [tex]\((x^2)(2x^2 - 2x - 1) = 2x^4 - 2x^3 - x^2\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(2x^4 - 2x^3 - 19x^2 + 11x + 3) - (2x^4 - 2x^3 - x^2) = -18x^2 + 11x + 3
\][/tex]
### Step 4: Repeat the Process
- Take the new dividend [tex]\(-18x^2 + 11x + 3\)[/tex].
- Divide [tex]\(-18x^2\)[/tex] by the leading term of the divisor [tex]\(2x^2\)[/tex] to get [tex]\(-9\)[/tex].
- Multiply the divisor by [tex]\(-9\)[/tex] to get [tex]\((-9)(2x^2 - 2x - 1) = -18x^2 + 18x + 9\)[/tex].
- Subtract this result from the current dividend:
[tex]\[
(-18x^2 + 11x + 3) - (-18x^2 + 18x + 9) = -7x - 6
\][/tex]
### Step 5: Conclusion
The division leads to a quotient of [tex]\(x^2 - 9\)[/tex] and a remainder of [tex]\(-7x - 6\)[/tex].
### Result
So, the result of dividing [tex]\( (2x^4 - 2x^3 - 19x^2 + 11x + 3) \)[/tex] by [tex]\( (2x^2 - 2x - 1) \)[/tex] is:
[tex]\[
\text{Quotient: } x^2 - 9, \quad \text{Remainder: } -7x - 6
\][/tex]
This means:
[tex]\[
2x^4 - 2x^3 - 19x^2 + 11x + 3 = (2x^2 - 2x - 1)(x^2 - 9) + (-7x - 6)
\][/tex]
### Step 1: Set Up the Division
Write the dividend ([tex]\(2x^4 - 2x^3 - 19x^2 + 11x + 3\)[/tex]) and the divisor ([tex]\(2x^2 - 2x - 1\)[/tex]) for polynomial long division.
### Step 2: Divide the Leading Terms
- Look at the leading term of the dividend [tex]\(2x^4\)[/tex] and the leading term of the divisor [tex]\(2x^2\)[/tex].
- Divide [tex]\(2x^4\)[/tex] by [tex]\(2x^2\)[/tex] to get the first term of the quotient, [tex]\(x^2\)[/tex].
### Step 3: Multiply and Subtract
- Multiply the entire divisor [tex]\(2x^2 - 2x - 1\)[/tex] by the term just found, [tex]\(x^2\)[/tex], to get [tex]\((x^2)(2x^2 - 2x - 1) = 2x^4 - 2x^3 - x^2\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(2x^4 - 2x^3 - 19x^2 + 11x + 3) - (2x^4 - 2x^3 - x^2) = -18x^2 + 11x + 3
\][/tex]
### Step 4: Repeat the Process
- Take the new dividend [tex]\(-18x^2 + 11x + 3\)[/tex].
- Divide [tex]\(-18x^2\)[/tex] by the leading term of the divisor [tex]\(2x^2\)[/tex] to get [tex]\(-9\)[/tex].
- Multiply the divisor by [tex]\(-9\)[/tex] to get [tex]\((-9)(2x^2 - 2x - 1) = -18x^2 + 18x + 9\)[/tex].
- Subtract this result from the current dividend:
[tex]\[
(-18x^2 + 11x + 3) - (-18x^2 + 18x + 9) = -7x - 6
\][/tex]
### Step 5: Conclusion
The division leads to a quotient of [tex]\(x^2 - 9\)[/tex] and a remainder of [tex]\(-7x - 6\)[/tex].
### Result
So, the result of dividing [tex]\( (2x^4 - 2x^3 - 19x^2 + 11x + 3) \)[/tex] by [tex]\( (2x^2 - 2x - 1) \)[/tex] is:
[tex]\[
\text{Quotient: } x^2 - 9, \quad \text{Remainder: } -7x - 6
\][/tex]
This means:
[tex]\[
2x^4 - 2x^3 - 19x^2 + 11x + 3 = (2x^2 - 2x - 1)(x^2 - 9) + (-7x - 6)
\][/tex]
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