Question 226·Hard·Equivalent Expressions

If a binomial is written as $ax + b$, what value of x makes that expression equal 0? Find that x-value for each of I, II, and III.

Substitute each x-value you found into $24x^3 - 2x^2 - 55x + 25$. If the result is 0, the corresponding binomial is a factor.

Be sure to test every candidate binomial; more than one (or even all) can be factors of the same polynomial.

In Desmos, type f(x) = 24x^3 - 2x^2 - 55x + 25 to define the polynomial as a function.

Click the plus (+) icon and choose Table, then in the x-column enter the three values corresponding to the binomials: -5/3, 5/4, and 1/2. Look at the f(x) values for each row.

For each x-value where f(x) is exactly 0, the corresponding binomial (3x + 5, 4x − 5, or 2x − 1) is a factor of the polynomial. Use which ones give 0 to decide which Roman numerals to include in your answer choice.

A binomial $ax + b$ is a factor of a polynomial $P(x)$ if and only if $P(-b/a) = 0$.

So for each binomial:

• I. $3x + 5 = 0 \Rightarrow x = -5/3$
• II. $4x - 5 = 0 \Rightarrow x = 5/4$
• III. $2x - 1 = 0 \Rightarrow x = 1/2$

We will plug each of these x values into $P(x) = 24x^3 - 2x^2 - 55x + 25$.

Compute $P(-5/3)$.

Work term by term:

• $\left(-\frac{5}{3}\right)^3 = -\frac{125}{27}$, so $24\cdot -\frac{125}{27} = -\frac{3000}{27} = -\frac{1000}{9}$
• $\left(-\frac{5}{3}\right)^2 = \frac{25}{9}$, so $-2 \cdot \frac{25}{9} = -\frac{50}{9}$
• $-55\cdot -\frac{5}{3} = \frac{275}{3} = \frac{825}{9}$
• $25 = \frac{225}{9}$

Now sum with denominator 9:

Thus $P(-5/3) = 0$.

Compute $P(5/4)$.

Work term by term:

• $\left(\frac{5}{4}\right)^3 = \frac{125}{64}$, so $24\cdot \frac{125}{64} = \frac{3000}{64} = \frac{375}{8}$
• $\left(\frac{5}{4}\right)^2 = \frac{25}{16}$, so $-2 \cdot \frac{25}{16} = -\frac{50}{16} = -\frac{25}{8}$
• $-55\cdot \frac{5}{4} = -\frac{275}{4}$

Combine the first two terms with denominator 8:

Now include the third term and 25:

So $P(5/4) = 0$ as well.

Compute $P(1/2)$.

Work term by term:

• $\left(\frac{1}{2}\right)^3 = \frac{1}{8}$, so $24\cdot \frac{1}{8} = 3$
• $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$, so $-2\cdot \frac{1}{4} = -\frac{1}{2}$
• $-55\cdot \frac{1}{2} = -\frac{55}{2}$

Group terms:

Thus $P(1/2) = 0$.

We found that

• $P(-5/3) = 0$, so $3x + 5$ (I) is a factor.
• $P(5/4) = 0$, so $4x - 5$ (II) is a factor.
• $P(1/2) = 0$, so $2x - 1$ (III) is a factor.

Therefore all three binomials I, II, and III are factors of $24x^3 - 2x^2 - 55x + 25$, so the correct answer choice is I, II, and III (D).