Question

factories p(z)=z³+13z²+32z+20​

Answer 1

Factorise the given cubic polynomial:

→ z³ + 13z² + 32z + 20

Here, constant term = 20

Factors of 20 = ±1 , ±2 , ±4 , ±5, ±10 , ±20

By trial and error method let us find out the first factor of the given polynomial.

• z - 1

→ z - 1 = 0

→ z = 0 + 1

→ z = 1

~ Put z as 1 in the given polynomial.

→ z³ + 13z² + 32z + 20

→ (1)³ + 13(1)² + 32(1) + 20

→ 1 + 13 + 32 + 20

→ 14 + 32 + 20

→ 14 + 52

→ 66 ≠ 0

• z + 1

→ z - 1 = 0

→ z = 0 - 1

→ z = -1

~ Put z as -1 in the given polynomial.

→ z³ + 13z² + 32z + 20

→ (-1)³ + 13(-1)² + 32(-1) + 20

→ -1 + 13 - 32 + 20

→ +12 - 32 + 20

→ +12 - 12

→ 0

• Therefore, first factor of the given polynomial = (z+1)

Now let us divide p(z) by g(z)

→ Kindly refer to attachment 1st and refer to attachment 2nd for the working note.

How to divide!? Use law of exponents perfectly and the rules to solve this type of questions! :)

...Therefore, we get z² + 12z + 20 as the quotient.

Now let us use middle term splitting method on the quotient and let us find out the other two factors.

→ z² + 12z + 20

• 2 × 10 = 20 ; 10 + 2 = 12

→ z² + 10z + 2z + 20

→ z(z+10) + 2(z+10)

→ (z+2)(z+10)

• (z+2)(z+10) are the other two factors of the given polynomial. $\:$