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find the hcf of:
[tex](3a {}^{2} + 5a - 12)(2a { \\ }^{2} + 7a + 3)(a {}^{2} + 4a + 3)[/tex]
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Sagot :

Hi1315

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).

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