Understanding Factors and the Factor Theorem

The question asks us to find the value of 'k' given that \((x - 2)\) is a factor of the polynomial \((2x^2 + 12kx - 25k)\).

In algebra, a factor of a polynomial is an expression that divides the polynomial exactly, leaving no remainder. This concept is strongly related to the roots or zeros of the polynomial.

The key principle we use here is the Factor Theorem. The Factor Theorem states:

• An expression \((x - a)\) is a factor of a polynomial \(P(x)\) if and only if \(P(a) = 0\). In other words, if you substitute the root of the factor into the polynomial, the result is zero.

In our specific problem, the given factor is \((x - 2)\). According to the Factor Theorem, if \((x - 2)\) is a factor of the polynomial \(P(x) = 2x^2 + 12kx - 25k\), then substituting \(x = 2\) into the polynomial must make the polynomial equal to zero.

First, we find the root of the factor \((x - 2)\) by setting it to zero:

• \(x - 2 = 0\)
• \(x = 2\)

Now, we substitute this value of \(x = 2\) into the given polynomial \((2x^2 + 12kx - 25k)\) and set the expression equal to zero.

Step-by-Step Solution to Find the Value of k

Let \(P(x) = 2x^2 + 12kx - 25k\).

Since \((x - 2)\) is a factor, according to the Factor Theorem, \(P(2) = 0\).

Substitute \(x = 2\) into the polynomial:

\(P(2) = 2(2)^2 + 12k(2) - 25k\)

Now, set \(P(2)\) equal to 0 and solve for \(k\):

\(2(2)^2 + 12k(2) - 25k = 0\)

Simplify the expression:

\(2(4) + 24k - 25k = 0\)

\(8 + (24k - 25k) = 0\)

Combine the terms with \(k\):

\(8 - k = 0\)

Isolate \(k\) by adding \(k\) to both sides of the equation:

\(8 = k\)

So, the value of k is 8.

Let's summarize the substitution step:

Setting the sum to zero: \(8 - k = 0\), which gives \(k = 8\).

Therefore, the value of k that makes \((x - 2)\) a factor of \((2x^2 + 12kx - 25k)\) is 8.

Revision Table: Key Concepts

Additional Information: Factor Theorem and Remainder Theorem

The Factor Theorem is actually a special case of the Remainder Theorem.

• Remainder Theorem: When a polynomial \(P(x)\) is divided by \((x - a)\), the remainder is \(P(a)\).
• Factor Theorem: If the remainder \(P(a)\) is 0 when \(P(x)\) is divided by \((x - a)\), then \((x - a)\) is a factor of \(P(x)\). This is because a zero remainder means the division is exact.

Understanding the relationship between factors, roots (or zeros), and these theorems is fundamental to working with polynomials. A root 'a' of a polynomial \(P(x)\) means \(P(a) = 0\), which directly implies that \((x - a)\) is a factor.

In this problem, finding the value of k relied entirely on the application of the Factor Theorem by setting the polynomial equal to zero at the root of the given factor \((x - 2)\).