DIRECTIONS: Factor each trinomial completely The word or phrase next to the equivalent expression w complete the statement correctly if it cannot be factored, the answer is "not factorable" $$ 3 d^{2}-4 d-7 $$ Yunus was offered a Fulbnght Scholarship and earned a Ph D. in economics at $$ \begin{array}{l}\text { (3d + 1)(d - 7) Belmont College } \ \text { (d - 1) }(3 d+7) \text { Loyola University } \ \text { (d }+1)(3 d-7) \text { Vanderbit University }\end{array} $$

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UpStudy AI · Accepted Answer

1. Write the trinomial:  
   $$
   3d^2 - 4d - 7
   $$

2. Compute the discriminant $$ \Delta $$ using  
   $$
   \Delta = b^2 - 4ac,
   $$  
   where $$ a = 3 $$, $$ b = -4 $$, and $$ c = -7 $$.  
   $$
   \Delta = (-4)^2 - 4(3)(-7) = 16 + 84 = 100
   $$

3. Since $$ \Delta = 100 $$ is a perfect square, the trinomial can be factored over the integers.

4. Find the roots using the quadratic formula:  
   $$
   d = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{4 \pm 10}{6}
   $$
   - For the positive square root:  
     $$
     d = \frac{4 + 10}{6} = \frac{14}{6} = \frac{7}{3}
     $$
   - For the negative square root:  
     $$
     d = \frac{4 - 10}{6} = \frac{-6}{6} = -1
     $$

5. Express the factors in terms of $$ d $$. The roots indicate the solutions of  
   $$
   d - \frac{7}{3} = 0 \quad \text{and} \quad d + 1 = 0.
   $$
   Multiply the factor with the fractional coefficient by 3 to clear the fraction:
   $$
   3\left(d - \frac{7}{3}\right) = 3d - 7.
   $$

6. Therefore, the factorization of the trinomial is:  
   $$
   (3d-7)(d+1)
   $$

7. Comparing with the given options, the equivalent expression is:  
   $$
   (d+1)(3d-7)
   $$
The trinomial $$ 3d^2 - 4d - 7 $$ factors to $$ (d + 1)(3d - 7) $$.

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