Please help me find the answer to this problem. I have to find k and a. Please tell me the method?

Question

If the polynomial x^4 - 6x^3 + 16x^2 - 25x + 10 is divided by x^2 - 2x + k, the remainder is x + a. Find k and a.

Answer 1

First, we divide the polynomial using long division:

................. x² - 4x + (8-k)
x² - 2x + k|x⁴ - 6x³ + 16x² - 25x + 10
................. x⁴ - 2x³ + kx²
................. -------------------------------
...................... -4x³ + (16-k)x² -25x
...................... -4x³ + 8x² ....... -4kx
...................... ---------------------------
...................... ......... (8-k)x² - (25-4k)x + 10
..................... .......... (8-k)x² - (16-2k)x + (8k-k²)
..................... .......... ------------------------------------
..................... ....................... - (9-2k)x + 10-8k+k²

So the remainder is - (9-2k)x + 10-8k+k². Since we know the remainder is x+a, equating coefficients yields 1=-(9-2k) and a=10-8k-k². Solving the first equation yields k=5, and substituting that into the second equation yields a = 10-40+25 = -5

Check of solution: if we have done our computations correctly, then multiplying our quotient with the divisor and adding the remainder should yields the original dividend. So:

(x² - 2x + k) * (x² - 4x + (8-k)) + x + a
(x² - 2x + 5) * (x² - 4x + 3) + x - 5
(x⁴ - 2x³ + 5x² - 4x³ + 8x² - 20x + 3x² - 6x + 15) + x-5
x⁴ - 6x³ + 16x² - 26x + 15 + x - 5
x⁴ - 6x³ + 16x² - 25x + 10

Which is the original polynomial. So our calculations were correct, and we are done. k=5, a=-5.

Answer 2

Hi there,

From the given expressions, we have the following equation:

x^4 - 6x^3 + 16x^2 - 25x + 10 = P(x)*(x^2 - 2x + k) + (x + a)

Note that the left hand side is a polynomial of degree 4, and so the expression P(x)*(x^2 - 2x + k) should also be of degree 4. Therefore, we deduce that P(x) must be a polynomial of degree 2 or a quadratic trinomial.

Hence, we let P(x) = x^2 + nx + m for some unknown values n and m. Notice that x^2 term has a coefficient 1, since the coefficient of x^2 in (x^2 - 2x + k) is also 1 and the coefficient of x^4 in the expression (x^4 - 6x^3 + 16x^2 - 25x + 10) is also 1, which we deduced from comparing coefficients.

So we have:

x^4 - 6x^3 + 16x^2 - 25x + 10 = P(x)*(x^2 - 2x + k) + (x + a)

=>

x^4 - 6x^3 + 16x^2 - 25x + 10 = (x^2 + nx + m)*(x^2 - 2x + k) + (x + a)

=>

x^4 - 6x^3 + 16x^2 - 25x + 10 = x^4 + (n-2)x^3 + (m-2n+k)x^2 + (kn -2m + 1)x + (mk + a).

Next, we compare the coefficients of each term on both sides:

For x^3:
-6 = n-2

For x^2:
16 = m - 2n + k

For x:
-25 = kn -2m + 1

For 1:
10 = mk + a.

For x^3, we readily get n = -4. Hence, the other equations will reduce to:

[Eq#1]
16 = m - 2n + k >> 16 = m +8+k >> 8 = m + k.

[Eq#2]
-25 = kn -2m + 1 >> -26 = -4k - 2m >> 13 = 2k + m

[Eq#3]
10 = mk + a.

If we subtract Eq#1 from Eq#2, we readily get: k = 13-8 = 5.

Substituting k=5 back to either Eq#1 or E#2 we get m = 3.

Finally, substituting k=5 and m=3 to Eq#3, we have:

10 = mk + a
>>
10 = 3*5 + a
>>
a = -5

Looking back at the problem, thus we found k and a:

k = 5,
a = -5.

:)

Answer 3

x^4 - 6x^3 + 16x^2 - 25x + 10 = (x^2+Cx+D)(x^2 - 2x + k) + (x+a)

so,

x^4 - 6x^3 + 16x^2 - 25x + 10 = x^4 + x^3(C-2) + x^2(k+D-2C) + x(Ck-2D+1) + (Dk+a)

you can the equate coefficients (or use an equivalent method)

x^3 term: C=-4
x^2 term: 16=k+D-2C, so, k+D = 8
x term: -25 = Ck-2D+1, so, 2k+D=13 , so, k=5 and D=3
constant term: 10=Dk+a, so, a=-5

answer: k = 5, a = -5