Answer :
The remainder when [tex]4x^4 - 10x^3 + 14x^2 + 7x - 19[/tex] is divided by 2x − 1 is -13.
The given expressions are [tex]4x^4 - 10x^3 + 14x^2 + 7x - 19[/tex] and 2x-1.
What is the remainder?
The Remainder is the value left after the division. If a number (dividend) is not completely divisible by another number (divisor) then we are left with a value once the division is done. This value is called the remainder.
The division of the given expressions are given below:
2x³-4x²+5x+6
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2x-1 | [tex]4x^4 - 10x^3 + 14x^2 + 7x - 19[/tex]
- ([tex]4x^4-2x^3[/tex])
____________________
-8x³+14x²+7x-19.
-(-8x³+4x²)
______________________
10x²+7x-19
-(10x²-5x)
__________
12x-19
-(12x-6)
_________
-13
Therefore, the remainder when [tex]4x^4 - 10x^3 + 14x^2 + 7x - 19[/tex] is divided by 2x − 1 is -13.
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Using long division, we have
___________________
2x-1 | 4x^4-10x³+14x²+7x-19
Next, we divide the first term of 2x-1 by the first term of 4x^4-10x³+14x²+7x-19 to get 4x^4/2x=2x³ (since 4/2=2 and to divide exponents you subtract the denominator from the numerator). Next, we put our 2x³ on the top and multiply (2x-1) by that. With the result of (2x-1) and 2x³, we multiply that by -1 and add it to 4x^4-10x³+14x²+7x-19, looking like
2x³
___________________
2x-1 | 4x^4-10x³+14x²+7x-19
- (4x^4-2x³)
____________________
-8x³+14x²+7x-19. Repeating the process, we get
2x³-4x²+5x+6
___________________
2x-1 | 4x^4-10x³+14x²+7x-19
- (4x^4-2x³)
____________________
-8x³+14x²+7x-19.
-(-8x³+4x²)
______________________
10x²+7x-19
-(10x²-5x)
__________
12x-19
-(12x-6)
_________
-13 as our remainder
___________________
2x-1 | 4x^4-10x³+14x²+7x-19
Next, we divide the first term of 2x-1 by the first term of 4x^4-10x³+14x²+7x-19 to get 4x^4/2x=2x³ (since 4/2=2 and to divide exponents you subtract the denominator from the numerator). Next, we put our 2x³ on the top and multiply (2x-1) by that. With the result of (2x-1) and 2x³, we multiply that by -1 and add it to 4x^4-10x³+14x²+7x-19, looking like
2x³
___________________
2x-1 | 4x^4-10x³+14x²+7x-19
- (4x^4-2x³)
____________________
-8x³+14x²+7x-19. Repeating the process, we get
2x³-4x²+5x+6
___________________
2x-1 | 4x^4-10x³+14x²+7x-19
- (4x^4-2x³)
____________________
-8x³+14x²+7x-19.
-(-8x³+4x²)
______________________
10x²+7x-19
-(10x²-5x)
__________
12x-19
-(12x-6)
_________
-13 as our remainder
