find the value of p for which (x-1 ) is a factor of x³+10x²+px
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Answer:
The value of p is - 11.
Step-by-step-explanation:
The given polynomial is x³ + 10x² + px.
Let the polynomial be P ( x ).
We have given that,
( x - 1 ) is a factor of the given polynomial.
By factor theorem,
If x = 1, P ( x ) = 0
∴ ( 1 )³ + 10 * ( 1 )² + p * 1 = 0
⇒ 1 + 10 * 1 + p = 0
⇒ 1 + 10 + p = 0
⇒ 11 + p = 0
⇒ p = - 11
∴ The value of p is - 11.
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Verification:
The given polynomial is x³ + 10x² + px.
We have, p = - 11.
By substituting p = - 11 in polynomial,
P ( x ) = x³ + 10x² - 11x
⇒ P ( x ) = x ( x² + 10x - 11 )
⇒ P ( x ) = x ( x² - x + 11x - 11 )
⇒ P ( x ) = x [ x ( x - 1 ) + 11 ( x - 1 ) ]
⇒ P ( x ) = x [ ( x - 1 ) ( x + 11 ) ]
⇒ P ( x ) = x ( x - 1 ) ( x + 11 )
( x - 1 ) is a factor of the polynomial x³ + 10x² + px when p = - 11.
Hence verified!
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