Finding K When X^3-7x^2+kx+15 Is Divisible By (x-3)

In this article, we will delve into the problem of finding the value of 'k' in the polynomial expression $x^3 - 7x^2 + kx + 15$ given that it is exactly divisible by $(x-3)$. This problem is a classic example of applying the Factor Theorem and polynomial division in algebra. Understanding these concepts is crucial for solving various problems in mathematics, especially those involving polynomials and their roots.

Understanding the Factor Theorem

The Factor Theorem is a fundamental concept in algebra that links the roots of a polynomial to its factors. It states that for a polynomial $f(x)$, if $f(a) = 0$ for some value 'a', then $(x-a)$ is a factor of $f(x)$. Conversely, if $(x-a)$ is a factor of $f(x)$, then $f(a) = 0$. This theorem provides a powerful tool for finding factors and roots of polynomials.

In simpler terms, if substituting a value 'a' into a polynomial results in zero, it means that $(x-a)$ divides the polynomial evenly, leaving no remainder. This is the core principle we will use to solve our problem.

Applying the Factor Theorem to the Problem

Given that the polynomial $x^3 - 7x^2 + kx + 15$ is exactly divisible by $(x-3)$, we can apply the Factor Theorem. This means that if we substitute $x = 3$ into the polynomial, the result should be zero. Mathematically, this can be written as:

$f(3) = (3)^3 - 7(3)^2 + k(3) + 15 = 0$

Now, we can solve this equation for 'k'. This involves simplifying the expression and isolating 'k' on one side of the equation.

Solving for k

Let's break down the equation step-by-step:

1. Calculate the powers:

   $(3)^3 = 27$ and $(3)^2 = 9$

2. Substitute these values back into the equation:

   $27 - 7(9) + 3k + 15 = 0$

3. Perform the multiplication:

   $27 - 63 + 3k + 15 = 0$

4. Combine the constant terms:

   $27 - 63 + 15 = -21$

5. Rewrite the equation:

   $-21 + 3k = 0$

6. Isolate the term with 'k':

   $3k = 21$

7. Solve for 'k':

   $k = \frac{21}{3} = 7$

Therefore, the value of k that makes the polynomial exactly divisible by $(x-3)$ is 7.

Verification

To verify our result, we can substitute $k = 7$ back into the original polynomial and perform polynomial division by $(x-3)$. If the remainder is zero, our solution is correct.

Substituting $k = 7$, the polynomial becomes:

$x^3 - 7x^2 + 7x + 15$

Now, let's perform polynomial long division:

 x^2 - 4x - 5
x - 3 | x^3 - 7x^2 + 7x + 15
       -(x^3 - 3x^2)
       -----------------
            -4x^2 + 7x
            -(-4x^2 + 12x)
            -----------------
                  -5x + 15
                  -(-5x + 15)
                  -------------
                       0

As we can see, the remainder is 0, which confirms that $(x-3)$ is indeed a factor of the polynomial when $k = 7$. This verification step is crucial to ensure the accuracy of our solution.

Alternative Approach: Polynomial Long Division

While the Factor Theorem provides a direct method to solve this problem, we can also use polynomial long division to find the value of 'k'. This method involves dividing the polynomial $x^3 - 7x^2 + kx + 15$ by $(x-3)$ and setting the remainder to zero.

The process of polynomial long division is as follows:

1. Divide the first term of the dividend ($x^3$) by the first term of the divisor ($x$), which gives $x^2$.
2. Multiply the divisor $(x-3)$ by $x^2$, resulting in $x^3 - 3x^2$.
3. Subtract this result from the dividend: $(x^3 - 7x^2) - (x^3 - 3x^2) = -4x^2$.
4. Bring down the next term from the dividend, which is $+kx$. The new expression is $-4x^2 + kx$.
5. Divide the first term of the new expression $(-4x^2)$ by the first term of the divisor ($x$), which gives $-4x$.
6. Multiply the divisor $(x-3)$ by $-4x$, resulting in $-4x^2 + 12x$.
7. Subtract this result from the new expression: $(-4x^2 + kx) - (-4x^2 + 12x) = (k - 12)x$.
8. Bring down the last term from the dividend, which is $+15$. The new expression is $(k - 12)x + 15$.
9. Divide the first term of the new expression $(k - 12)x$ by the first term of the divisor ($x$), which gives $(k - 12)$.
10. Multiply the divisor $(x-3)$ by $(k - 12)$, resulting in $(k - 12)x - 3(k - 12)$.
11. Subtract this result from the new expression: $((k - 12)x + 15) - ((k - 12)x - 3(k - 12)) = 15 + 3(k - 12)$.

For the polynomial to be exactly divisible by $(x-3)$, the remainder must be zero. Therefore, we set the remainder equal to zero:

$15 + 3(k - 12) = 0$

Now, we can solve this equation for 'k':

1. Distribute the 3:

   $15 + 3k - 36 = 0$

2. Combine the constant terms:

   $3k - 21 = 0$

3. Isolate the term with 'k':

   $3k = 21$

4. Solve for 'k':

   $k = \frac{21}{3} = 7$

This approach also yields the same result, $k = 7$, reinforcing the correctness of our solution. Polynomial long division is a versatile method for dividing polynomials and can be used in various algebraic problems.

Conclusion

In conclusion, we have successfully found the value of 'k' in the polynomial $x^3 - 7x^2 + kx + 15$ that makes it exactly divisible by $(x-3)$. We used two methods: the Factor Theorem and polynomial long division. Both methods led to the same result, $k = 7$. Understanding and applying these methods are essential for solving problems related to polynomials and their factors.

The Factor Theorem provides a quick and efficient way to find factors and roots of polynomials, while polynomial long division is a more general method that can be used to divide polynomials of any degree. By mastering these techniques, you can confidently tackle a wide range of algebraic problems.

Therefore, the correct answer to the question "If $x^3-7x^2+kx+15$ is exactly divisible by $(x-3)$, then k equals" is:

A. 7