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Question
If 3 and –3 are two zeroes of the polynomial (x4 + x3 – 11x2 – 9x + 18), find all the zeroes of the given polynomial.
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Solution
Let x4 + x3 – 11x2 – 9x + 18
Since 3 and –3 are the zeroes of f(x), it follows that each one of (x + 3) and (x – 3) is a factor of f(x).
Consequently, (x – 3) (x + 3) = (x2 – 9) is a factor of f(x).
On dividing f(x) by (x2 – 9), we get:
`x^2 - 9")"overline(x^4 + x^3 - 11x^2 - 9x + 18)"("x^2 + x - 2`
x4 – 9x2
– +
x3 – 2x2 – 9x + 18
x3 – 9x
– +
–2x2 + 18
–2x2 + 18
+ –
x
f(x) = 0 ⇒ (x2 + x – 2) (x2 – 9) = 0
⇒ (x2 + 2x – x – 2) (x – 3) (x + 3)
⇒ (x – 1) (x + 2) (x – 3) (x + 3) = 0
⇒ x = 1 or x = –2 or x = 3 or x = –3
Hence, all the zeroes are 1, –2, 3 and –3.
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