The polynomial 3x3 + hx2 - 5x + k is denoted by p(x). Given that (3x + 2) and (x- 1)
are factors of p(x), find the values of h and ofk. Hence factorise the polyomial completely.
Since (3x + 2) and (x - 1) are factors of the polynomial p(x), we can set the polynomial equal to zero and solve for x.
Setting p(x) = 0, we have:
(3x + 2)(x - 1) = 0
Expanding this expression, we get:
3x^2 - 3x + 2x - 2 = 0
Combining like terms, we have:
3x^2 - x - 2 = 0
Comparing this equation to the given polynomial p(x) = 3x^3 + hx^2 - 5x + k, we can equate the corresponding coefficients:
h = -1 (coefficient of x^2 term)
k = -2 (constant term)
Therefore, the values of h and k are -1 and -2, respectively.
To factorize the polynomial completely, we use the values of h and k to rewrite p(x) as:
p(x) = 3x^3 - x^2 - 5x - 2
Now we can see that p(x) has a common factor of (x - 1) and (3x + 2). Factoring out these common factors, we have:
p(x) = (x - 1)(3x + 2)(x + 2)
Hence, the polynomial is completely factorized as (x - 1)(3x + 2)(x + 2).
Given that (3x + 2) and (x - 1) are factors of p(x), we can write p(x) as:
p(x) = (3x + 2)(x - 1)(ax + b)
To find the values of h and k, we can expand p(x) and compare the coefficients with the given polynomial 3x^3 + hx^2 - 5x + k:
p(x) = (3x + 2)(x - 1)(ax + b)
= (3x + 2)(ax^2 + bx - ax - b)
= (3x^2 + 2x - 3ax - 2a)(ax + b)
= 3ax^3 + 2ax^2 - 3ax^2 - 2a x + 3ax + 2x - 3ax - 2a
= 3ax^3 - ax^2 + 2x - 2a
Comparing the coefficients, we have:
3a = 3 (coefficient of x^3)
-a = h (coefficient of x^2)
2 = -5 (coefficient of x)
-2a = k (constant term)
From the first equation, we find a = 1.
From the second equation, we find -a = h, so h = -1.
From the third equation, we find 2 = -5, which is not possible.
Thus, there is no value for k that satisfies all the equations.
Factoring the polynomial completely using the found values of a and h:
p(x) = 3ax^3 - ax^2 + 2x - 2a
= 3x^3 - x^2 + 2x + 2
Therefore, the factored form of the polynomial is:
p(x) = (3x + 2)(x - 1)(x + 1)
To find the values of h and k, we can use the fact that (3x + 2) and (x - 1) are factors of p(x).
When (3x + 2) is a factor of p(x), it means that p(-2/3) = 0. Let's substitute this into the polynomial equation:
p(-2/3) = 3(-2/3)^3 + h(-2/3)^2 - 5(-2/3) + k
= -8/3 + 4h/9 + 10/3 + k
Since p(-2/3) = 0, we have:
-8/3 + 4h/9 + 10/3 + k = 0
Simplifying this equation, we get:
-8 + 4h + 30 + 9k = 0
4h + 9k = -22 (Equation 1)
Similarly, when (x - 1) is a factor of p(x), it means that p(1) = 0. Let's substitute this into the polynomial equation:
p(1) = 3(1)^3 + h(1)^2 - 5(1) + k
= 3 + h - 5 + k
Since p(1) = 0, we have:
3 + h - 5 + k = 0
Simplifying this equation, we get:
h + k - 2 = 0
h + k = 2 (Equation 2)
Now we have a system of equations consisting of Equation 1 and Equation 2. Let's solve this system to find the values of h and k.
Solving the system of equations (Equation 1 and Equation 2) we have:
4h + 9k = -22
h + k = 2
We can use any method to solve this system of linear equations, such as substitution or elimination. Let's solve it by elimination:
Multiply the second equation by 4:
4(h + k) = 4(2)
4h + 4k = 8 (Equation 3)
Now subtract Equation 3 from Equation 1:
(4h + 9k) - (4h + 4k) = -22 - 8
5k = -30
k = -6
Substituting the value of k into Equation 2:
h + (-6) = 2
h - 6 = 2
h = 8
Therefore, the values of h and k are h = 8 and k = -6.
Now that we found the values of h and k, let's factorize the polynomial completely using the given factors:
We know that (3x + 2) and (x - 1) are factors of p(x). So, we can express p(x) as the product of these factors:
p(x) = (3x + 2)(x - 1)
To factorize it completely, we need to check if there are any more factors. This can be done by finding the roots of the polynomial equation p(x) = 0.
Setting p(x) = 0:
(3x + 2)(x - 1) = 0
This equation will be true if either (3x + 2) = 0 or (x - 1) = 0:
For (3x + 2) = 0, we have:
3x + 2 = 0
3x = -2
x = -2/3
For (x - 1) = 0, we have:
x - 1 = 0
x = 1
Therefore, the roots of the polynomial are x = -2/3 and x = 1.
So, the complete factorization of the polynomial p(x) = 3x^3 + hx^2 - 5x + k is:
p(x) = (3x + 2)(x - 1)(x + 2/3)
I hope this helps! Let me know if you have any further questions.