Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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]
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]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.