Question

which is a factor of $72x^5 - 200x^3$?
a $4x - 5$
b $5x + 3$
c $3x - 5$
d $2x + 3$

Explanation:

Step1: Factor the polynomial

First, factor out the greatest common factor (GCF) from \(72x^5 - 200x^3\). The GCF of \(72x^5\) and \(200x^3\) is \(8x^3\). So we have:
$$72x^5 - 200x^3 = 8x^3(9x^2 - 25)$$

Step2: Apply difference of squares

Notice that \(9x^2 - 25\) is a difference of squares, since \(9x^2=(3x)^2\) and \(25 = 5^2\). The formula for the difference of squares is \(a^2 - b^2=(a + b)(a - b)\). Here, \(a = 3x\) and \(b = 5\), so:
$$9x^2 - 25=(3x + 5)(3x - 5)$$

Step3: Combine the factors

Putting it all together, the factored form of \(72x^5 - 200x^3\) is:
$$72x^5 - 200x^3 = 8x^3(3x + 5)(3x - 5)$$
Now we check the options:

• Option A: \(4x - 5\) is not a factor.
• Option B: \(5x + 3\) is not a factor.
• Option C: \(3x - 5\) is a factor (from the factored form above).
• Option D: \(2x + 3\) is not a factor.

Answer:

C. \(3x - 5\)