Write each polynomial below as a product of first degree polynomials. Write also the solutions of the equation p(x) = 0 in each.

p(x) = x² + 7x + 12

Given, p(X) = x² + 7x + 12

Now, we need to write the given polynomial as a product of first degree polynomial and p(x) = 0

⇒ x² + 7x + 12 = 0

⇒ the given equation can be written as follows

⇒ (x + 4)(x + 3) = 0

⇒ Since, we know that x² + 7x + 12 can be written as (x + a)(x + b)

⇒ x² + 7x + 12 = x² + (a + b)x + ab

∴ coefficient on either side of the polynomial is same and a + b = 7 and ab = 12

⇒ we must find two numbers that satisfy a + b and ab we get, 4 and 3 as the numbers

∴ (x + 4)(x + 3) are the product of first degree polynomial

⇒ P(x) = 0 if (x + 3) is 0 and (x + 4) is 0

∴ x + 4 = 0

⇒ x = – 4

⇒ x² + 7x + 12 = (– 4)² + 7(– 4) + 12 = 16 – 28 + 12 = 0

And x + 3 = 0

⇒ x = – 3

⇒ x² + 7x + 12 = (– 3)² + 7(– 3) + 12 = 9 – 21 + 12 = 0

Hence, (x + 4)(x + 3) are the first degree factors of the polynomial and – 4, – 3 are the solutions of the given polynomial x² + 7x + 12