Question

Find all the roots of following question:–

m(x) = x³ + 5x² – 24x

Final Answer should be:–

x = 0
x = 3
x= –8

Solve step by step.
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Answer 1

Question:

Find all the roots of the following polynomial :

➼ m (x) = x³ + 5x² – 24x

Answer:

The roots of the polynomial m (x) = x³ + 5x² – 24x are :

➼ x

➼ (x – 3)

➼ (x + 8)

Step-by-step explanation:

Factorising x³ + 5x² – 24x by removing the common term 'x' :

➼ x³ + 5x² – 24x = 0

➼ x (x² + 5x – 24) = 0

So, we have two factors :

➼ x = 0

➼ x² + 5x – 24 = 0

Now, let's split the middle term of x² + 5x – 24 and factorise :

➼ x² + 5x – 24 = 0

➼ x² + 8x – 3x – 24 = 0

➼ x (x + 8) – 3 (x + 8) = 0

➼ (x + 8) (x – 3) = 0

➼ x = – 8 and x = 3

Verification:

Let's substitute x = 0 firstly :

➼ x³ + 5x² – 24x = 0

➼ ( 0 )³ + 5 ( 0 )² – 24 ( 0 ) = 0

➼ 0 + 0 + 0 = 0

➼ 0 = 0

➼ LHS = RHS

Substituting x = 3 now :

➼ x³ + 5x² – 24x = 0

➼ ( 3 )³ + 5 ( 3 )² – 24 ( 3 ) = 0

➼ 27 + 5 ( 9 ) – 72 = 0

➼ 27 + 45 – 72 = 0

➼ 72 – 72 = 0

➼ 0 = 0

➼ LHS = RHS

Substituting x = – 8 finally :

➼ x³ + 5x² – 24x = 0

➼ (– 8 )³ + 5 (– 8 )² – 24 (– 8 ) = 0

➼ – 512 + 5 ( 64 ) + 192 = 0

➼ – 320 + 320 = 0

➼ 0 = 0

➼ LHS = RHS

Hence, verifed!