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प्रश्न
Prove that (5x + 4) is a factor of 5x3 + 4x2 – 5x – 4. Hence factorize the given polynomial completely.
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उत्तर
f(x) = 5x3 + 4x2 – 5x – 4
Let 5x + 4 = 0, then 5x = -4
⇒ x = `(-4)/(5)`
∴ `f(-4/5) = 5(-4/5)^3 + 4(-4/5)^2 -5(-4/5) -4`
= `5 xx (-64/125) + 4 xx (16)/(25) + 4 - 4`
= `-(64)/(25) + (64)/(25) + 4 - 4` = 0
∵ `f(-4/5)` = 0
∴ (5x + 4) is a factor of f(x)
Now dividing f(x) by 5x + 4, we get
5x3 + 4x2 – 5x – 4
= (5x + 4)(x2 – 1)
= (5x + 4)[(x)2 – (1)2]
= (5x + 4)(x + 1)(x – 1)
`5x + 4")"overline(5x^3 + 4x^2 - 5x - 4)("x^2 - 1`
5x3 + 4x2
– –
–5x – 4
–5x – 4
+ +
x
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