Question

When \(3 x^{2}+6 x-10\) is divided by \(x+k\) the remainder is \(14 .\) Determine the value(s) of \(k\)

Short answer

k = 4 or k = -2.

Step-by-step solution

- Understand Remainder Theorem

The Remainder Theorem states that for a polynomial function f(x), the remainder of the division of f(x) by (x-r) is f(r). Here, you are given the polynomial f(x) = 3x^2 + 6x - 10 and the divisor (x+k).

- Express Division Form

Rewrite the divisor in the form (x - r). Since the divisor is given as (x + k), it can be rewritten as (x - (-k)). Therefore, r = -k.

- Apply Remainder Theorem

According to the Remainder Theorem, f(-k) = 14. Substitute -k into the polynomial: 3(-k)^2 + 6(-k) - 10 = 14.

- Simplify the Equation

Simplify the equation: 3k^2 - 6k - 10 = 14.

- Solve for k

Move all terms to one side to form a quadratic equation: 3k^2 - 6k - 24 = 0. Divide the entire equation by 3: k^2 - 2k - 8 = 0. Factorize: (k - 4)(k + 2) = 0.

- Find Values of k

Set each factor to zero: k - 4 = 0 or k + 2 = 0. So, k = 4 or k = -2.

Key concepts

Polynomial Division

Polynomial division breaks down a complex polynomial into simpler terms. Think of it like long division you learned in elementary school, but with variables. To illustrate, let's use the given problem with the polynomial \(\f(x) = 3x^2 + 6x - 10\). When this polynomial is divided by \(x + k\), we're essentially asking how many times \(x + k\) fits into \(f(x)\) and what is left over. This 'leftover' is the remainder, which, according to the problem, is 14. The Remainder Theorem simplifies this process. It allows us to find that remainder without performing the entire division. Instead, we evaluate the polynomial at a specific point (in this case, at \(-k\)). Understanding this principle helps in grasping deeper algebraic concepts later.

Solving Quadratic Equations

To solve a quadratic equation means finding the values of the variable that make the equation true. For the given exercise, after applying the Remainder Theorem, we generated a quadratic equation: \(3k^2 - 6k - 24 = 0\). A quadratic equation is an equation where the highest exponent of the variable is 2. Here, \(3k^2 - 6k - 24 = 0\) is transformed into a simpler form, \(k^2 - 2k - 8 = 0\), by dividing all terms by 3. Solving involves isolating \k\. One method is factoring, where you express the equation as a product of binomials. In this case, it becomes \