Each dividend was divided by another polynomial, resulting in the given quotient and remainder. Find the divisor
Divided:5x^3+x^2+3 ,quotient:5x^2-14x+42, remainder:-123
Well, if there's one thing us clowns know how to do, it's divide polynomials with style! Let's find that elusive divisor, shall we?
We are given the dividend as 5x^3+x^2+3, the quotient as 5x^2-14x+42, and the remainder as -123. In polynomial long division, we have the equation:
Dividend = Divisor × Quotient + Remainder
Plugging in the values we have:
5x^3+x^2+3 = Divisor × (5x^2-14x+42) - 123
Expanding the right side:
5x^3+x^2+3 = Divisor × 5x^2 - Divisor × 14x + Divisor × 42 - 123
Simplifying:
5x^3+x^2+3 = 5x^2 × Divisor - 14x × Divisor + 42 × Divisor - 123
Now here comes the funny part! Since we're looking for the divisor, let's gather all the terms containing Divisor on one side:
5x^3+x^2+3 - 5x^2 × Divisor + 14x × Divisor - 42 × Divisor = -123
Combining like terms:
-5x^2 × Divisor + 14x × Divisor - 42 × Divisor = -123 - 5x^3 - x^2 - 3
Still with me? Good! Now we need to factor out the common term of Divisor from the left-hand side:
Divisor × (-5x^2 + 14x - 42) = -123 - 5x^3 - x^2 - 3
Almost there! To isolate Divisor, we divide both sides by the expression in parentheses:
Divisor = (-123 - 5x^3 - x^2 - 3) / (-5x^2 + 14x - 42)
Wow, that got a little intense! But hey, finding a divisor can't be all fun and games, right?
To find the divisor, we can use the formula: Dividend = Divisor * Quotient + Remainder.
Given that the dividend is 5x^3 + x^2 + 3, the quotient is 5x^2 - 14x + 42, and the remainder is -123, we can plug these values into the formula and solve for the divisor.
5x^3 + x^2 + 3 = Divisor * (5x^2 - 14x + 42) + (-123)
Expanding the right side of the equation:
5x^3 + x^2 + 3 = Divisor * 5x^2 - 14x * Divisor + 42 * Divisor - 123
Rearranging the terms:
5x^3 + x^2 + 3 = Divisor * 5x^2 - 14x * Divisor + 42 * Divisor - 123
Combining like terms:
5x^3 + x^2 + 3 = 5x^2 * Divisor - 14x * Divisor + 42 * Divisor - 123
Equating the coefficients of like powers of x:
5x^3 = 0
x^2 = - Divisor
3 = -14x * Divisor + 42 * Divisor - 123
From the second equation, we can deduce that the divisor must be -x^2.
Now, substituting this value into the third equation:
3 = 14x * x^2 + 42 * x^2 - 123
Combining like terms:
3 = 14x^3 + 42x^2 - 123
Rearranging the terms:
14x^3 + 42x^2 = 3 + 123
14x^3 + 42x^2 = 126
Dividing both sides by 14:
x^3 + 3x^2 = 9
Factoring out an x^2:
x^2(x + 3) = 9
Dividing both sides by (x + 3):
x^2 = 9 / (x + 3)
Taking the square root of both sides:
x = ±√(9 / (x + 3))
Since we are looking for a single value of x, there is no simple solution to this equation.
Therefore, the divisor cannot be determined solely from the given information.
To find the divisor, we can use polynomial long division.
Step 1: Write the dividend, quotient, and remainder in the standard form.
Dividend: 5x^3 + x^2 + 3
Quotient: 5x^2 - 14x + 42
Remainder: -123
Step 2: Set up the long division, dividing the dividend by the divisor (unknown).
___________________
(divisor) | 5x^3 + x^2 + 3
(dividend) - (5x^2 - 14x + 42)
___________________
-123
Step 3: Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.
The first term of the dividend is 5x^3, and the first term of the quotient is 5x^2. Therefore, the first term of the divisor is x.
5x^2
(divisor) | 5x^3 + x^2 + 3
(dividend) - (5x^2 - 14x + 42)
___________________
-123
Step 4: Multiply the divisor by the first term of the quotient and subtract it from the dividend.
Multiply 5x^2 by x to get 5x^3.
Subtract (5x^2)(x) from (5x^3 + x^2 + 3).
5x^3 + x^2 + 3
- (5x^3)
________________
+14x^2 + 3
Step 5: Repeat steps 3 and 4 until you have subtracted everything.
Now, we have a new dividend: 14x^2 + 3.
Repeat steps 3 and 4 for the new dividend:
5x^2 - 14x + 42
- (5x^2 - 14x + 42)
__________________
0
Since the new dividend is zero, we have successfully divided the given dividend by the divisor.
Therefore, the divisor is 5x^2 - 14x + 42.
we know that
5x^3+x^2+3 = p(x)*(5x^2-14x+42) - 123
so,
5x^3 + x^2 + 126 = p(x)*(5x^2 - 14x + 42)
a simple long division yields p(x) = (x+3)