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 of [tex]\(p(x)\)[/tex] by [tex]\((x+2)\)[/tex].
### Step-by-step Solution:
1. Setup the Division:
We divide [tex]\(p(x)\)[/tex] by [tex]\((x+2)\)[/tex]. In polynomial long division, we look at the leading term of the dividend and the divisor. The leading term of [tex]\(p(x)\)[/tex] is [tex]\(25x^4\)[/tex] and the leading term of the divisor is [tex]\(x\)[/tex].
2. First Term of Quotient:
Divide [tex]\(25x^4\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(25x^3\)[/tex]. Multiply [tex]\((x+2)\)[/tex] by [tex]\(25x^3\)[/tex] to get [tex]\(25x^4 + 50x^3\)[/tex].
3. Subtract to Find the New Dividend:
Subtract [tex]\(25x^4 + 50x^3\)[/tex] from the original polynomial:
[tex]\[
(25x^4 + 70x^3 - 11x^2 - 84x + 36) - (25x^4 + 50x^3) = 20x^3 - 11x^2 - 84x + 36
\][/tex]
4. Second Term of Quotient:
Divide [tex]\(20x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(20x^2\)[/tex]. Multiply [tex]\((x+2)\)[/tex] by [tex]\(20x^2\)[/tex] to get [tex]\(20x^3 + 40x^2\)[/tex].
5. Subtract to Get the Next Dividend:
[tex]\[
(20x^3 - 11x^2 - 84x + 36) - (20x^3 + 40x^2) = -51x^2 - 84x + 36
\][/tex]
6. Third Term of Quotient:
Divide [tex]\(-51x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-51x\)[/tex]. Multiply [tex]\((x+2)\)[/tex] by [tex]\(-51x\)[/tex] to get [tex]\(-51x^2 - 102x\)[/tex].
7. Subtract to Find Remaining Dividend:
[tex]\[
(-51x^2 - 84x + 36) - (-51x^2 - 102x) = 18x + 36
\][/tex]
8. Fourth Term of Quotient:
Divide [tex]\(18x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(18\)[/tex]. Multiply [tex]\((x+2)\)[/tex] by [tex]\(18\)[/tex] to get [tex]\(18x + 36\)[/tex].
9. Subtract to Find Remainder:
[tex]\[
(18x + 36) - (18x + 36) = 0
\][/tex]
Since the remainder is [tex]\(0\)[/tex], [tex]\((x+2)\)[/tex] is indeed a factor of [tex]\(p(x)\)[/tex].
10. Final Result:
The quotient from the division, which is the remaining factors of [tex]\(p(x)\)[/tex] after dividing by [tex]\((x+2)\)[/tex], is [tex]\(25x^3 + 20x^2 - 51x + 18\)[/tex]. Therefore, the polynomial [tex]\(p(x)\)[/tex] can be written as:
[tex]\[
p(x) = (x+2)(25x^3 + 20x^2 - 51x + 18)
\][/tex]
Thus, [tex]\((x+2)\)[/tex] is a factor, and the remaining factor, after factoring out [tex]\((x+2)\)[/tex], is [tex]\(25x^3 + 20x^2 - 51x + 18\)[/tex].
### Step-by-step Solution:
1. Setup the Division:
We divide [tex]\(p(x)\)[/tex] by [tex]\((x+2)\)[/tex]. In polynomial long division, we look at the leading term of the dividend and the divisor. The leading term of [tex]\(p(x)\)[/tex] is [tex]\(25x^4\)[/tex] and the leading term of the divisor is [tex]\(x\)[/tex].
2. First Term of Quotient:
Divide [tex]\(25x^4\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(25x^3\)[/tex]. Multiply [tex]\((x+2)\)[/tex] by [tex]\(25x^3\)[/tex] to get [tex]\(25x^4 + 50x^3\)[/tex].
3. Subtract to Find the New Dividend:
Subtract [tex]\(25x^4 + 50x^3\)[/tex] from the original polynomial:
[tex]\[
(25x^4 + 70x^3 - 11x^2 - 84x + 36) - (25x^4 + 50x^3) = 20x^3 - 11x^2 - 84x + 36
\][/tex]
4. Second Term of Quotient:
Divide [tex]\(20x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(20x^2\)[/tex]. Multiply [tex]\((x+2)\)[/tex] by [tex]\(20x^2\)[/tex] to get [tex]\(20x^3 + 40x^2\)[/tex].
5. Subtract to Get the Next Dividend:
[tex]\[
(20x^3 - 11x^2 - 84x + 36) - (20x^3 + 40x^2) = -51x^2 - 84x + 36
\][/tex]
6. Third Term of Quotient:
Divide [tex]\(-51x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-51x\)[/tex]. Multiply [tex]\((x+2)\)[/tex] by [tex]\(-51x\)[/tex] to get [tex]\(-51x^2 - 102x\)[/tex].
7. Subtract to Find Remaining Dividend:
[tex]\[
(-51x^2 - 84x + 36) - (-51x^2 - 102x) = 18x + 36
\][/tex]
8. Fourth Term of Quotient:
Divide [tex]\(18x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(18\)[/tex]. Multiply [tex]\((x+2)\)[/tex] by [tex]\(18\)[/tex] to get [tex]\(18x + 36\)[/tex].
9. Subtract to Find Remainder:
[tex]\[
(18x + 36) - (18x + 36) = 0
\][/tex]
Since the remainder is [tex]\(0\)[/tex], [tex]\((x+2)\)[/tex] is indeed a factor of [tex]\(p(x)\)[/tex].
10. Final Result:
The quotient from the division, which is the remaining factors of [tex]\(p(x)\)[/tex] after dividing by [tex]\((x+2)\)[/tex], is [tex]\(25x^3 + 20x^2 - 51x + 18\)[/tex]. Therefore, the polynomial [tex]\(p(x)\)[/tex] can be written as:
[tex]\[
p(x) = (x+2)(25x^3 + 20x^2 - 51x + 18)
\][/tex]
Thus, [tex]\((x+2)\)[/tex] is a factor, and the remaining factor, after factoring out [tex]\((x+2)\)[/tex], is [tex]\(25x^3 + 20x^2 - 51x + 18\)[/tex].
