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Question
Using Remainder Theorem, factorise : x3 + 10x2 – 37x + 26 completely.
Solution
By remainder theorem,
For x = 1, the value of the given expression is the remainder.
x3 + 10x2 – 37x + 26
= (1)3 + 10(1)2 – 37(1) + 26
= 1 + 10 – 37 + 26
= 37 – 37
= 0
x – 1 is a factor of x3 + 10x2 – 37x + 26.
x2 + 11x – 26
`x - 1")"overline(x^3 + 10x^2 - 37x + 26)`
x3 – x2
11x2 – 37x
11x2 – 11x
– 26x + 26
– 26x + 26
0
∴ x3 + 10x2 – 37x + 26
= (x – 1)(x2 + 11x – 26)
= (x – 1)(x2 + 13x – 2x – 26)
= (x – 1)[x(x + 13) – 2(x + 13)]
= (x – 1)(x + 13)(x – 2)
∴ x3 + 10x2 – 37 + 26 = (x – 1)(x + 13)(x – 2)
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