First, let's examine the given problem. We are asked to determine what must be subtracted from the polynomial [tex]\( p(x) = 4x^4 - 2x^3 - 6x^2 - x - 5 \)[/tex] so that the result is divisible by the polynomial [tex]\( d(x) = 2x^2 + x - 1 \)[/tex].
To solve this problem, we will use polynomial division to find the quotient and the remainder when [tex]\( p(x) \)[/tex] is divided by [tex]\( d(x) \)[/tex].
1. Identify the polynomials:
- Dividend ( [tex]\( p(x) \)[/tex] ): [tex]\( 4x^4 - 2x^3 - 6x^2 - x - 5 \)[/tex]
- Divisor ( [tex]\( d(x) \)[/tex] ): [tex]\( 2x^2 + x - 1 \)[/tex]
2. Polynomial Division:
When we divide [tex]\( p(x) \)[/tex] by [tex]\( d(x) \)[/tex], we obtain a quotient and a remainder. These can be represented as:
[tex]\[
p(x) = (d(x) \cdot q(x)) + r(x)
\][/tex]
where [tex]\( q(x) \)[/tex] is the quotient and [tex]\( r(x) \)[/tex] is the remainder.
3. Given Result:
From the previous computation, we know the result of the division:
[tex]\[
\text{Quotient } q(x) = 2x^2 - 2x - 1
\][/tex]
[tex]\[
\text{Remainder } r(x) = -2x - 6
\][/tex]
4. Formulate the relationship:
The division gives us:
[tex]\[
p(x) = (d(x) \cdot q(x)) + r(x)
\][/tex]
Plugging in the values, we have:
[tex]\[
4x^4 - 2x^3 - 6x^2 - x - 5 = (2x^2 + x - 1)(2x^2 - 2x - 1) + (-2x - 6)
\][/tex]
5. Determine what must be subtracted:
So, to make [tex]\( p(x) \)[/tex] divisible by [tex]\( d(x) \)[/tex], we need the remainder to be zero. This means we need to subtract the remainder [tex]\( -2x - 6 \)[/tex] from the polynomial [tex]\( p(x) \)[/tex].
Thus, the expression that must be subtracted from [tex]\( 4x^4 - 2x^3 - 6x^2 - x - 5 \)[/tex] to make it divisible by [tex]\( 2x^2 + x - 1 \)[/tex] is [tex]\( -2x - 6 \)[/tex].
In conclusion:
[tex]\[
\boxed{-2x - 6}
\][/tex]
