Questions: Factor the trinomial by grouping. 8 x^2 + 14 x + 3 a. Find two numbers whose product is 8 * 3 = 24 and whose sum is 14. b. Write 14 x using the factors from part (a). c. Factor by grouping. a. The two numbers with a product of 24 and a sum of 14 are

Factor the trinomial by grouping.
8 x^2 + 14 x + 3
a. Find two numbers whose product is 8 * 3 = 24 and whose sum is 14.
b. Write 14 x using the factors from part (a).
c. Factor by grouping.
a. The two numbers with a product of 24 and a sum of 14 are

Transcript text: Factor the trinomial by grouping. \[ 8 x^{2}+14 x+3 \] a. Find two numbers whose product is $8 \cdot 3=24$ and whose sum is 14 . b. Write 14 x using the factors from part (a). c. Factor by grouping. a. The two numbers with a product of 24 and a sum of 14 are $\square$ (Use a comma to separate answers as needed.)

Solution

Solution Steps

To factor the trinomial $8x^2 + 14x + 3$ by grouping, follow these steps:

Find two numbers whose product is $8 \cdot 3 = 24$ and whose sum is 14.
Use these two numbers to split the middle term $14x$ into two terms.
Factor by grouping the resulting four-term polynomial.

Step 1: Find the Two Numbers

To factor the trinomial $8x^2 + 14x + 3$, we first need to find two numbers that multiply to $8 \cdot 3 = 24$ and add up to $14$. The numbers that satisfy these conditions are $12$ and $2$.

Step 2: Rewrite the Middle Term

Next, we rewrite the middle term $14x$ using the two numbers found in Step 1: \[ 8x^2 + 12x + 2x + 3 \]

Step 3: Factor by Grouping

Now, we group the terms: \[ (8x^2 + 12x) + (2x + 3) \] Factoring out the common factors in each group gives us: \[ 4x(2x + 3) + 1(2x + 3) \] Now, we can factor out the common binomial $(2x + 3)$: \[ (2x + 3)(4x + 1) \]