Finding Factors of a Quadratic Polynomial

The question asks us to find a factor of the quadratic polynomial \(f(x) = 2x^2 - 5x + 2\) from the given options.

A common way to check if a linear expression \(x - c\) is a factor of a polynomial \(f(x)\) is by using the Factor Theorem. The Factor Theorem states that \(x - c\) is a factor of \(f(x)\) if and only if \(f(c) = 0\).

We can test each option by substituting the corresponding value of \(c\) into the function \(f(x)\) and checking if the result is zero.

• Option A: \(x - 2\)
• This means \(c = 2\). We need to evaluate \(f(2)\).

Let's calculate \(f(2)\):

\(f(2) = 2(2)^2 - 5(2) + 2\)

\(f(2) = 2(4) - 10 + 2\)

\(f(2) = 8 - 10 + 2\)

\(f(2) = -2 + 2\)

\(f(2) = 0\)

Since \(f(2) = 0\), according to the Factor Theorem, \(x - 2\) is a factor of \(f(x) = 2x^2 - 5x + 2\).

Let's quickly check the other options to confirm (although finding one factor is enough to answer the question):

• Option B: \(x - 3\)
• This means \(c = 3\). Evaluate \(f(3)\).

\(f(3) = 2(3)^2 - 5(3) + 2\)

\(f(3) = 2(9) - 15 + 2\)

\(f(3) = 18 - 15 + 2\)

\(f(3) = 3 + 2\)

\(f(3) = 5\)

Since \(f(3) \neq 0\), \(x - 3\) is not a factor.

• Option C: \(x - 4\)
• This means \(c = 4\). Evaluate \(f(4)\).

\(f(4) = 2(4)^2 - 5(4) + 2\)

\(f(4) = 2(16) - 20 + 2\)

\(f(4) = 32 - 20 + 2\)

\(f(4) = 12 + 2\)

\(f(4) = 14\)

Since \(f(4) \neq 0\), \(x - 4\) is not a factor.

• Option D: \(x - 5\)
• This means \(c = 5\). Evaluate \(f(5)\).

\(f(5) = 2(5)^2 - 5(5) + 2\)

\(f(5) = 2(25) - 25 + 2\)

\(f(5) = 50 - 25 + 2\)

\(f(5) = 25 + 2\)

\(f(5) = 27\)

Since \(f(5) \neq 0\), \(x - 5\) is not a factor.

Only for Option A (\(x - 2\)) did the function evaluate to 0, confirming it is a factor.

Revision Table: Testing Factors

Additional Information: Factoring Quadratics

Besides testing factors, we can also factor the quadratic expression \(2x^2 - 5x + 2\) directly.

We look for two numbers that multiply to \(2 \times 2 = 4\) and add up to \(-5\). These numbers are \(-4\) and \(-1\).

We can rewrite the middle term \(-5x\) as \(-4x - x\):

\(2x^2 - 5x + 2 = 2x^2 - 4x - x + 2\)

Now, group the terms and factor by grouping:

\((2x^2 - 4x) + (-x + 2)\)

Factor out common terms from each group:

\(2x(x - 2) - 1(x - 2)\)

Now factor out the common binomial factor \((x - 2)\):

\((x - 2)(2x - 1)\)

The factors of \(2x^2 - 5x + 2\) are \((x - 2)\) and \((2x - 1)\). Our direct factoring confirms that \(x - 2\) is indeed a factor of the polynomial \(f(x)\).