If the binomial (a – 5) is a factor of the polynomial a² + 7a + 10, what is the other factor?

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To determine the other factor of the polynomial a² + 7a + 10 given that (a – 5) is one of its factors, we can perform polynomial division or use the factorization method.

First, we can express the polynomial as follows, knowing one of its factors:

a² + 7a + 10 = (a – 5)(?)

To find the other factor, we can set up an equation:

Let (a – 5)(A) be equal to a² + 7a + 10. We need to find A.

We can expand (a – 5)(A):

(a – 5)(A) = aA – 5A

Setting this equal to the original polynomial gives us:

aA – 5A = a² + 7a + 10

Now we will factor the original polynomial through simple observation or trial and error. The numbers that multiply to give 10 and add up to 7 are 5 and 2. Therefore, we can factor the polynomial as follows:

a² + 7a + 10 = (a + 5)(a + 2)

Since we were initially given that (a – 5) is a factor, we can try the polynomial division method:

Using synthetic division or polynomial long division:

1. Divide a² + 7a + 10 by (a - 5):

  1. Step 1: How many times does a go into a²? 1 time.
  2. Step 2: Multiply (a - 5) by 1: gives a - 5.
  3. Step 3: Subtract this from (a² + 7a + 10):
  - (a² - 5 a) = 7a + 5a + 10 = 12a + 10.
  4. Repeat until fully divided.

By continuing this process, we can find the other factor to be (a + 2).

In conclusion, given that (a – 5) is a factor of the polynomial a² + 7a + 10, the other factor is:

(a + 2).

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