what must be subtracted from 4x4 - 2x3 - 6x2 + x - 5 so that the resulting polynomial is exactly divisible by 2x2 +3x-2
Answers
The remainder you get should be subtracted from polynomial so that the new polynomial obtained is divisible by 2x2 + 3x - 2
Answer:
-22x+5 should be subtracted so that 4x⁴-2x³-6x²+x-5 is exactly divisible by 2x²+3x-2
Step-by-step explanation:
To find,
The expression to be subtracted from 4x⁴-2x³-6x²+x-5 so that the resulting polynomial is exactly divisible by 2x²+3x-2
Let f(x) be 4x⁴-2x³-6x²+x-5 and g(x) be 2x²+3x-2
First, let us divide the polynomial f(x) by g(x)
2x²-4x+5
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2x²+3x-2 I 4x⁴-2x³-6x²+x-5
4x⁴+6x³-x²
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-8x³-2x²+x-5
-8x³-12x²+8x
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+10x² - 7x-5
+10x² +15x-10
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-22x+5
Then by division algorithm, we have,
Dividend = Quotient x Divisor + Remainder
f(x) = g(x) x Quotient + Remainder.
f(x) - Remainder = g(x) x Quotient.
f(x) - Remainder is a multiple of g(x)
f(x) is exactly divided by g(x), when the remainder is subtracted from f(x)
Here, dividend = f(x) = 4x⁴-2x³-6x²+x-5
Quotient = g(x) = 2x²-4x+5
Divisor = 2x²+3x-2
Remainder = -22x+5
Then we have f(x) - ( -22x+5) is exactly divisible by g(x)
∴ -22x+5 should be subtracted so that 4x⁴-2x³-6x²+x-5 is exactly divisible by 2x²+3x-2
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