Answer :
To determine if [tex]\((x + 2)\)[/tex] is a factor of the polynomial [tex]\(p(x) = 25x^4 + 70x^3 - 11x^2 - 84x + 36\)[/tex], we perform polynomial division to see if dividing [tex]\(p(x)\)[/tex] by [tex]\((x + 2)\)[/tex] gives a remainder of zero.
Here's the step-by-step process:
1. Set up the division: We want to divide the polynomial [tex]\(p(x)\)[/tex] by [tex]\((x + 2)\)[/tex].
2. Perform the division:
- Divide the leading term of [tex]\(p(x)\)[/tex], which is [tex]\(25x^4\)[/tex], by the leading term of [tex]\((x + 2)\)[/tex], which is [tex]\(x\)[/tex]. This gives [tex]\(25x^3\)[/tex].
- Multiply [tex]\(25x^3\)[/tex] by [tex]\((x + 2)\)[/tex] to get [tex]\(25x^4 + 50x^3\)[/tex].
- Subtract [tex]\(25x^4 + 50x^3\)[/tex] from [tex]\(p(x)\)[/tex] to get a new polynomial: [tex]\(20x^3 - 11x^2 - 84x + 36\)[/tex].
3. Repeat the process:
- Divide the new leading term, [tex]\(20x^3\)[/tex], by [tex]\(x\)[/tex] to get [tex]\(20x^2\)[/tex].
- Multiply [tex]\(20x^2\)[/tex] by [tex]\((x + 2)\)[/tex] to get [tex]\(20x^3 + 40x^2\)[/tex].
- Subtract [tex]\(20x^3 + 40x^2\)[/tex] from the new polynomial to get: [tex]\(-51x^2 - 84x + 36\)[/tex].
4. Continue dividing:
- Divide the leading term, [tex]\(-51x^2\)[/tex], by [tex]\(x\)[/tex] to get [tex]\(-51x\)[/tex].
- Multiply [tex]\(-51x\)[/tex] by [tex]\((x + 2)\)[/tex] to get [tex]\(-51x^2 - 102x\)[/tex].
- Subtract [tex]\(-51x^2 - 102x\)[/tex] from the polynomial to get: [tex]\(18x + 36\)[/tex].
5. Final division:
- Divide the leading term, [tex]\(18x\)[/tex], by [tex]\(x\)[/tex] to get [tex]\(18\)[/tex].
- Multiply [tex]\(18\)[/tex] by [tex]\((x + 2)\)[/tex] to get [tex]\(18x + 36\)[/tex].
- Subtract [tex]\(18x + 36\)[/tex] from [tex]\(18x + 36\)[/tex] to get a remainder of [tex]\(0\)[/tex].
Since the remainder is [tex]\(0\)[/tex], [tex]\((x + 2)\)[/tex] is indeed a factor of [tex]\(p(x)\)[/tex].
The quotient obtained from this division is [tex]\(25x^3 + 20x^2 - 51x + 18\)[/tex].
Now, the remaining factors of [tex]\(p(x)\)[/tex] can be found. Since [tex]\((x + 2)\)[/tex] is confirmed as a factor, [tex]\(p(x)\)[/tex] can be expressed as:
[tex]\[ p(x) = (x + 2)(25x^3 + 20x^2 - 51x + 18) \][/tex]
The task is complete with this factorization, and we found that [tex]\((x + 2)\)[/tex] is indeed a factor of [tex]\(p(x)\)[/tex], with the remaining polynomial being [tex]\(25x^3 + 20x^2 - 51x + 18\)[/tex].
Here's the step-by-step process:
1. Set up the division: We want to divide the polynomial [tex]\(p(x)\)[/tex] by [tex]\((x + 2)\)[/tex].
2. Perform the division:
- Divide the leading term of [tex]\(p(x)\)[/tex], which is [tex]\(25x^4\)[/tex], by the leading term of [tex]\((x + 2)\)[/tex], which is [tex]\(x\)[/tex]. This gives [tex]\(25x^3\)[/tex].
- Multiply [tex]\(25x^3\)[/tex] by [tex]\((x + 2)\)[/tex] to get [tex]\(25x^4 + 50x^3\)[/tex].
- Subtract [tex]\(25x^4 + 50x^3\)[/tex] from [tex]\(p(x)\)[/tex] to get a new polynomial: [tex]\(20x^3 - 11x^2 - 84x + 36\)[/tex].
3. Repeat the process:
- Divide the new leading term, [tex]\(20x^3\)[/tex], by [tex]\(x\)[/tex] to get [tex]\(20x^2\)[/tex].
- Multiply [tex]\(20x^2\)[/tex] by [tex]\((x + 2)\)[/tex] to get [tex]\(20x^3 + 40x^2\)[/tex].
- Subtract [tex]\(20x^3 + 40x^2\)[/tex] from the new polynomial to get: [tex]\(-51x^2 - 84x + 36\)[/tex].
4. Continue dividing:
- Divide the leading term, [tex]\(-51x^2\)[/tex], by [tex]\(x\)[/tex] to get [tex]\(-51x\)[/tex].
- Multiply [tex]\(-51x\)[/tex] by [tex]\((x + 2)\)[/tex] to get [tex]\(-51x^2 - 102x\)[/tex].
- Subtract [tex]\(-51x^2 - 102x\)[/tex] from the polynomial to get: [tex]\(18x + 36\)[/tex].
5. Final division:
- Divide the leading term, [tex]\(18x\)[/tex], by [tex]\(x\)[/tex] to get [tex]\(18\)[/tex].
- Multiply [tex]\(18\)[/tex] by [tex]\((x + 2)\)[/tex] to get [tex]\(18x + 36\)[/tex].
- Subtract [tex]\(18x + 36\)[/tex] from [tex]\(18x + 36\)[/tex] to get a remainder of [tex]\(0\)[/tex].
Since the remainder is [tex]\(0\)[/tex], [tex]\((x + 2)\)[/tex] is indeed a factor of [tex]\(p(x)\)[/tex].
The quotient obtained from this division is [tex]\(25x^3 + 20x^2 - 51x + 18\)[/tex].
Now, the remaining factors of [tex]\(p(x)\)[/tex] can be found. Since [tex]\((x + 2)\)[/tex] is confirmed as a factor, [tex]\(p(x)\)[/tex] can be expressed as:
[tex]\[ p(x) = (x + 2)(25x^3 + 20x^2 - 51x + 18) \][/tex]
The task is complete with this factorization, and we found that [tex]\((x + 2)\)[/tex] is indeed a factor of [tex]\(p(x)\)[/tex], with the remaining polynomial being [tex]\(25x^3 + 20x^2 - 51x + 18\)[/tex].
