Question

question 3 (1 point)

which of the following is a factor of the polynomial function (h(x) = 2x^4 - 5x^3 - 9x^2 + 18x)?

• none of these
• all of these
• ((2x - 3))
• ((x - 3))
• (2x)
• ((x + 2))

Explanation:

⚡ Using what you learned: factoring completely

Step 1: Factor out the greatest common factor

Factor out \( x \) from the polynomial:

$$ h(x) = x(2x^3 - 5x^2 - 9x + 18) $$

Step 2: Factor by grouping

Group the terms inside the parentheses:

$$ 2x^3 - 5x^2 - 9x + 18 = x^2(2x - 5) - 9(2x - 5) $$

Factor out the common binomial \((2x - 5)\):

$$ (2x - 5)(x^2 - 9) $$

Step 3: Factor the difference of squares

Factor \( x^2 - 9 \):

$$ x^2 - 9 = (x - 3)(x + 3) $$

Combine all factors:

$$ h(x) = x(2x - 5)(x - 3)(x + 3) $$

Step 4: Identify the correct option

Comparing the factors \( x \), \( (2x - 5) \), \( (x - 3) \), and \( (x + 3) \) with the given choices:

• \( (2x - 3) \) is not a factor.
• \( (x - 3) \) is a factor.
• \( 2x \) is not a factor.
• \( (x + 2) \) is not a factor.

Answer:

\( (x - 3) \)