Answer:
(a+3)
Step-by-step explanation:
Given polynomials:
[tex]1. 3a^2 + 5a - 12\\\\2. 2a^2 + 7a + 3\\\\3. a^2 + 4a + 3\\\\[/tex]
We will factorize each polynomial to find their common factors.
Factorize the Polynomials
1. Factorize 3a² + 5a - 12
We need to find two numbers that multiply to 3 × -12 = -36 and add up to 5.
These numbers are 9 and -4:
3a² + 9a - 4a - 12
Now, factor by grouping:
3a(a + 3) - 4(a + 3)
= (3a - 4)(a + 3)
2. Factorize 2a² + 7a + 3
We need to find two numbers that multiply to 2 × 3 = 6 and add up to 7.
These numbers are 6 and 1:
2a² + 6a + a + 3
Now, factor by grouping:
2a(a + 3) + 1(a + 3)
= (2a + 1)(a + 3)
3. Factorize a² + 4a + 3
We need to find two numbers that multiply to 1 × 3 = 3 and add up to 4.
These numbers are 3 and 1:
a² + 3a + a + 3
Now, factor by grouping:
a(a + 3) + 1(a + 3)
= (a + 3)(a + 1)
Determine the HCF
The factorizations are:
1. 3a² + 5a - 12 = (3a - 4)(a + 3)
2. 2a² + 7a + 3 = (2a + 1)(a + 3)
3. a² + 4a + 3 = (a + 3)(a + 1)
The common factor among all three polynomials is (a + 3).
