Problem 11 - Beginning and Intermediate Algebra

Question

How do we know that \((2 x-4)\) cannot be a factor of \(2 x^{2}+13 x-24 ?\)

Step-by-Step Solution

Answer

After factoring out a common factor, we are left with \((x-2)\) as the simplified factor. Dividing the given polynomial, \(2x^2 + 13x - 24\), by \((x - 2)\) resulted in a quotient of \((2x + 5)\) and a non-zero remainder (10). Therefore, \((2x - 4)\), or equivalently \((x - 2)\), cannot be a factor of the given polynomial.

Step 1: Factor out common factor from (2x - 4)

We notice that both terms in \((2x-4)\) have a common factor of 2 that we can factor out, giving us: \[ (2x - 4) = 2( x - 2) \]

Step 2: Divide given polynomial by (x - 2)

Now, we will perform polynomial division to divide the given polynomial, \(2x^2 + 13x - 24\), by \((x - 2)\). Performing the division, we get: _______________(2x + 5)___ x - 2 | 2x^2+13x-24 -(2x^2- 4x)________ 17x-24 -(17x-34) _____________________ 10 So, the division yields a quotient of \((2x + 5)\) and a remainder of 10.

Step 3: Check the remainder

Since the remainder is non-zero (10), \((2x - 4)\) or equivalently \((x - 2)\) is not a factor of \((2x^2 + 13x - 24)\). Therefore, we know that \((2x - 4)\) cannot be a factor of the given expression.