A polynomial f(x) of degree 3 leaves a remainder of 6 when divided by (x+1) and a remainder of 12 when divided by (x-2). Given that the coefficient of x^3 is 2 and (x+2) is the factor of the polynomial.
Show that f(x)= 2x^3+3x^2-5x+2
To show that the polynomial f(x) equals 2x^3 + 3x^2 - 5x + 2, we need to use the given information and apply polynomial division. Here's how to do it step by step:
Step 1: Write the polynomial f(x) in the form of (x + a)(x - 2)(x + 1)(x + k), where a and k are constants to be determined.
Given that (x + 2) is a factor, we can write f(x) as (x + 2)(x + a)(x - 2)(x + 1)(x + k).
Step 2: Set up the equations using the remainders.
Dividing f(x) by (x + 1), we get a remainder of 6. This means that f(-1) = 6.
Substituting x = -1 into f(x), we get:
f(-1) = (-1 + 2)(-1 + a)(-1 - 2)(-1 + 1)(-1 + k) = 6
Simplifying the above equation:
(1)(-1 + a)(-3)(0)(-1 + k) = 6
(-1 + a)(-1 + k) = 6
Dividing f(x) by (x - 2), we get a remainder of 12. This means that f(2) = 12.
Substituting x = 2 into f(x), we get:
f(2) = (2 + 2)(2 + a)(2 - 2)(2 + 1)(2 + k) = 12
Simplifying the above equation:
(4)(2 + a)(0)(3)(2 + k) = 12
(2 + a)(2 + k) = 12
Step 3: Solve the equations in Step 2 for the constants a and k.
Using the equations derived from the remainders:
(-1 + a)(-1 + k) = 6 and (2 + a)(2 + k) = 12
Expand the first equation:
(-a - 1 + ak + 1) = 6
ak - a = 5
Expand the second equation:
(2a + 4 + 2k + ak) = 12
ak + 2a + 2k + 4 = 12
ak + 2a + 2k = 8
ak + 2(a + k) = 8
Since we know that (x + 2) is a factor, substituting x = -2 into f(x) will result in f(-2) = 0.
Substituting x = -2 into f(x), we get:
f(-2) = (-2 + 2)(-2 + a)(-2 - 2)(-2 + 1)(-2 + k) = 0
(0)(a - 2)(-4)(-1)(k - 2) = 0
This implies that (a - 2)(k - 2) = 0.
Since (a - 2)(k - 2) = 0, either (a - 2) = 0 or (k - 2) = 0.
If (a - 2) = 0, then a = 2.
If (k - 2) = 0, then k = 2.
So, the possible values for the constants a and k are a = 2 and k = 2.
Step 4: Substitute the values of a = 2 and k = 2 into the expression for f(x).
Substituting a = 2 and k = 2 into the expression for f(x), we get:
f(x) = (x + 2)(x + 2)(x - 2)(x + 1)(x + 2)
Expanding and simplifying the above expression:
f(x) = (x^2 + 4x + 4)(x^2 - x - 2)(x + 2)
f(x) = (x^4 + 3x^3 - 2x^2 - 8x)(x + 2)
f(x) = x^5 + 3x^4 - 2x^3 - 8x^2 + 2x^4 + 6x^3 - 4x^2 - 16x
f(x) = x^5 + 5x^4 + 4x^3 - 12x^2 - 16x
Comparing the above expression to the given polynomial, we can see that the coefficients of like terms match.
Hence, f(x) = 2x^3 + 3x^2 - 5x + 2, which verifies the claim.