The Correct Answer and Explanation is:
The correct factored form of the trinomial is (5x + 7)(x + 5).
Explanation:
To factor the trinomial 5x² + 32x + 35, we can use a method called factoring by grouping, which is often used for trin, at the same time, add up to the middle coefficient ‘b’, which is 32. Let’s list theomials in the form ax² + bx + c where a is not equal to 1.
factor pairs of 175 and check their sum:
- 1 and 175 (Sum = 176)
- 5 and 35 (Sum = 40)
- 7 andFirst, we identify the coefficients: a = 5, b = 32, and `c = 25 (Sum = 32)
The pair 7 and 25 meets both conditions. We 35`.
The initial step is to find the product of a and c. In this case, will use these two numbers to rewrite the middle term of the original trinomial, 32x, as the sum a * c = 5 * 35 = 175`.
Next, we need to find two numbers25x + 7x`.
The expression becomes:
5x² + 25x + 7 that both multiply to 175 and add up to the middle coefficient,b, which is 32x + 35
Next, we factor the expression by grouping the first two terms and the last two terms. We can list the factor pairs of 175 to find the correct combination:
- 1 and 175 (sum = 176)
- 5 and 35 (sum = 40:
(5x² + 25x) + (7x + 35)
Factor out the greatest common)
- 7 and 25 (sum = 32)
The pair 7 and 2 factor (GCF) from each group. The GCF of the first group is 5x, and the GCF of5 is the one we need because 7 * 25 = 175 and 7 + the second group is7.5x(x + 5) + 7(x + 25 = 32`.
Now, we rewrite the middle term of the original trinomial, 325)
Notice that both terms now share a common binomial factor, (x + 5). We can factor thisx, as the sum of two terms using the numbers we just found:7x + 25x. Our out.(5x + 7)(x + 5)`
To verify the answer, we can multiply the trinomial becomes:
5x² + 25x + 7x + 35
factors using the FOIL method:
- First: 5x * x = 5xThe next step is to factor by grouping. We group the first two terms and the last two terms:(5x²`
- Outer: 5x * 5 = 25x
*² + 25x) + (7x + 35)`
Then, we find the greatest common factor ( Inner: 7 * x = 7x
- Last:GCF) for each group and factor it out.
- For the first group, (5x²7 * 5 = 35Combining the terms gives5x² + 25x + + 25x), the GCF is5x. Factoring this out leaves us with5x 7x + 35 = 5x² + 32x + 35, which matches(x + 5).
- For the second group, (7x + 35), the original trinomial. the GCF is 7. Factoring this out gives us 7(x + 5).
Our expression is now 5x(x + 5) + 7(x + 5).
We can see that both terms now share a common binomial factor, (x + 5). We can factor this common binomial out, which gives us our final answer:
(x + 5)(5x + 7)
