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Question
If 1 and –2 are two zeroes of the polynomial (x3 – 4x2 – 7x + 10), find its third zero.
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Solution
Let f(x) = x3 – 4x2 – 7x + 10
Since 1 and –2 are the zeroes of f(x), it follows that each one of (x – 1) and (x + 2) is a factor of f(x).
Consequently, (x – 1) (x + 2) = (x2 + x – 2) is a factor of f(x).
On dividing f(x) by (x2 + x – 2), we get:
`x^2 + x - 2")"overline(x^3 - 4x^2 - 7x + 10)"("x - 5`
x3 + x2 – 2x
– – +
–5x2 – 5x + 10
–5x2 – 5x + 10
+ + –
X
f(x) = 0 ⇒ (x2 + x – 2) (x – 5) = 0
⇒ (x – 1) (x + 2) (x – 5) = 0
⇒ x = 1 or x = –2 or x = 5
Hence, the third zero is 5.
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