Factor completely 3x² + 5x + 1.
(3x + 1)(x + 1), (3x + 5)(x + 1), (3x - 5)(x + 1), Prime

Solution:

After visual inspection of the given quadratic equation i.e. 3x² +5x + 1 one can infer that it cannot be factorized.

The next step therefore will be to identify the nature of roots.

This can be done by using the formula to find the root of the quadratic equation of the form ax² + bx + c.

The formula is given as:

x = - b ± √(b² - 4ac)/ 2a --- (1)

The values of a, b and c for the given quadratic equation 3x² + x + 7 are as follows:

a = 3; b = 5; c = 1

The value of (b² - 4ac) can be calculated as:

5² - 4(3)(1) = 25 - 12 = 13

Since the number is a negative number its square root will be an imaginary number.

Hence the roots of the quadratic equation will be :

x = -5/6 ± (√13)/ 6

⇒ x is either -5/6 + (√13)/6 or -5/6 - (√13)/6

Hence it is verified that the given equation has factors which generate real and unequal roots.

One component of the root is rational i.e. -5/6 and other component is irrational i.e.

(√13)/6.

Hence, 3x² + 5x + 1 cannot be factorised.

Therefore it is prime.