When the polynomial
f(x)=(p-1)+x^3+px^2+qx+r, where p, q is divided by (x+2) and (x-1) the remainders are -5 and 4 respectively. If (x+1) is a factor of f(x), find the value of p, q and r, hence factorize f(x) completely.
To find the values of p, q, and r, we can use the Remainder Theorem. According to the theorem, if a polynomial f(x) is divided by (x - c), the remainder is equal to f(c).
Given that (x + 2) and (x - 1) are divisors of f(x) with remainders of -5 and 4 respectively, we can write the following equations:
f(-2) = -5 (since f(x) divided by (x + 2) gives a remainder of -5)
f(1) = 4 (since f(x) divided by (x - 1) gives a remainder of 4)
Now, let's substitute the given values into the equation f(x) = (p - 1) + x^3 + px^2 + qx + r:
f(-2) = (p - 1) + (-2)^3 + p(-2)^2 + q(-2) + r = -5
-8p - 2q + r + p - 1 = -5
-7p - 2q + r = -4 [Equation 1]
f(1) = (p - 1) + 1^3 + p(1)^2 + q(1) + r = 4
p + q + r = 4 [Equation 2]
To find the value of p, q, and r, solve the system of equations (Equation 1 and Equation 2).
Now let's solve the system of equations:
Equation 1: -7p - 2q + r = -4
Equation 2: p + q + r = 4
Adding the two equations, we get:
-6p - q + 2r = 0 [Equation 3]
Now, substituting Equation 3 into Equation 1, we get:
-7p - 2q + r = -4
-6p - q + 2r = 0
Simplifying Equation 3, we can write it as:
q = -6p + 2r
Substituting this back into Equation 1, we have:
-7p - 2(-6p + 2r) + r = -4
-7p + 12p - 4r + r = -4
5p - 3r = -4
To find the value of p and r, let's assume p = k (a variable), where k is any real number.
Now, substituting p = k into the equation above, we have:
5k - 3r = -4
Rearranging the equation, we can solve for r:
3r = 5k + 4
r = (5k + 4) / 3
We have found the value of r. Now, substitute this into the Equation 2 to find the value of p:
p + q + r = 4
p + q + (5k + 4) / 3 = 4
Simplify the equation:
3p + 3q + 5k + 4 = 12
3p + 3q = 12 - 5k - 4
3p + 3q = 8 - 5k
Now, let's assume q = m (a variable), where m is any real number.
Substitute q = m into the equation above and simplify:
3p + 3m = 8 - 5k
This equation can be written as:
3p = 8 - 3m - 5k
To find the value of p, let's assume m = 0:
3p = 8 - 5k
We can assume k = 0 as well:
3p = 8
Therefore, p = 8/3.
Substituting the values of p and r into Equation 2, we can solve for q:
p + q + r = 4
8/3 + q + (5k + 4) / 3 = 4
Multiply through by 3:
8 + 3q + 5k + 4 = 12
3q + 5k + 12 = 12 - 8
3q + 5k = -8
Now, assume k = 0:
3q = -8
Therefore, q = -8/3.
So, the values of p, q, and r are p = 8/3, q = -8/3, and r = (5k + 4) / 3.
Now, let's substitute these values back into the equation f(x)=(p-1) + x^3 + px^2 + qx + r to find f(x) completely:
f(x) = (8/3 - 1) + x^3 + (8/3)x^2 + (-8/3)x + (5k + 4)/3
f(x) = (5/3) + x^3 + (8/3)x^2 + (-8/3)x + (5k + 4)/3
Therefore, the factorization of f(x) is:
f(x) = (5/3) + x^3 + (8/3)x^2 + (-8/3)x + (5k + 4)/3
Please note that the expressions for p, q, and r depend on the variable k, which can be any real number.
To find the values of p, q, and r, we can use the Remainder Theorem. According to the Remainder Theorem, when a polynomial f(x) is divided by (x - a), the remainder is equal to f(a).
Given that the remainder when f(x) is divided by (x + 2) is -5, we can substitute -2 for x in the polynomial f(x) and set it equal to -5:
f(-2) = (-2 + 2)^3 + p(-2)^2 + q(-2) + r = -5
Simplifying this equation, we get:
0 + 4p - 2q + r = -5 ...(1)
Similarly, the remainder when f(x) is divided by (x - 1) is 4. Substituting 1 for x in f(x) and setting it equal to 4:
f(1) = (1 + 2)^3 + p(1)^2 + q(1) + r = 4
Simplifying this equation, we get:
27 + p + q + r = 4 ...(2)
Now, we know that (x + 1) is a factor of f(x), which means f(-1) = 0. Substituting -1 for x in f(x) and setting it equal to 0:
f(-1) = (-1 + 2)^3 + p(-1)^2 + q(-1) + r = 0
Simplifying this equation, we get:
1 - p - q + r = 0 ...(3)
We now have a system of three equations (equations 1, 2, and 3) with three variables (p, q, and r). Solving this system of equations will give us the values of p, q, and r.
To solve these equations, we can use any preferred method such as substitution or elimination. Let's use substitution:
From equation 3, we can express r in terms of p and q:
r = p + q - 1 ...(4)
Substituting this value of r in equations 1 and 2, we get:
4p - 2q + (p + q - 1) = -5 ...(5)
27 + p + q + (p + q - 1) = 4 ...(6)
Simplifying equations 5 and 6:
5p - q = -4 ...(7)
2p + 2q = -22 ...(8)
Now we have a system of two equations (equations 7 and 8) with two variables (p and q). Solving this system will give us the values of p and q:
Multiply equation 8 by 5:
10p + 10q = -110 ...(9)
Multiply equation 7 by 2:
10p - 2q = -8 ...(10)
Adding equations 9 and 10:
12q = -118
Simplifying, we find:
q = -118/12 = -59/6
Substituting this value of q into equation 7:
5p - (-59/6) = -4
5p + 59/6 = -4
Multiply by 6 to eliminate the fraction:
30p + 59 = -24
30p = -83
p = -83/30
Substituting the values of p and q back into equation 4 to find r:
r = (-83/30) + (-59/6) - 1
r = (-83 - 295 + 30)/30
r = -348/30 = -58/5
Hence, the values of p, q, and r are: p = -83/30, q = -59/6, and r = -58/5.
To factorize f(x) completely, we now have the values of p, q, and r:
f(x) = (p-1) + x^3 + px^2 + qx + r
= (-83/30 - 1) + x^3 + (-83/30)x^2 + (-59/6)x + (-58/5)
= (-113/30) + x^3 - (83/30)x^2 - (59/6)x - (58/5)
We can simplify this further, but the process is involved.
Not complete
from the given data:
f(x)=(p-1)+x^3+px^2+qx+r
f(1) = (p-1) + 1 + p + q + r = 4 ---> 2p + q + r = 4
f(-1) = (p-1) - 1 + p - q + r = 0 ----> 2p - q + r = 0
f(-2) = (p-1) - 8 + 4p - 2q + r = -5 --> 5p - 2q + r = -5
you now 3 equations in 3 unknowns, solve using your
favourite method, then factor the resulting f(x)