When a polynomial is divided by x+3, the quotient is x^3+x^2-4 and the remainder is 8. Find the dividend?
Suppose the question had said:
when a number is divided by 4 the quotient is 6 with a remainder of 1
we would write:
number = 6(4) + 1
= 25
so
f(x) = (x^3 + x^2 - 4)(x+3) + 8
= x^4 + 3x^3 + x^3 + 3x^2 - 4x - 12 + 8
= x^4 + 4x^3 + 3x^2 - 4x - 4
To find the dividend, we can use the division algorithm for polynomials. Let's set up the polynomial division:
Dividend / (x + 3) = Quotient + Remainder
Given that the quotient is x^3 + x^2 - 4 and the remainder is 8, we can rewrite the equation:
Dividend / (x + 3) = (x^3 + x^2 - 4) + 8
Now, let's subtract 8 from both sides of the equation:
Dividend / (x + 3) - 8 = x^3 + x^2 - 4
Next, let's multiply both sides of the equation by (x + 3) to eliminate the denominator:
Dividend - 8(x + 3) = (x^3 + x^2 - 4)(x + 3)
Expanding the right side of the equation:
Dividend - 8x - 24 = x^4 + 3x^3 + x^2 + 3x^2 + 9x - 4x - 12
Combining like terms:
Dividend - 8x - 24 = x^4 + 3x^3 + 4x^2 + 5x - 12
Now, let's bring all the terms to one side of the equation:
x^4 + 3x^3 + 4x^2 + 5x - 12 - Dividend + 8x + 24 = 0
Rearranging the terms:
x^4 + 3x^3 + 4x^2 + 13x + 12 - Dividend = 0
Finally, we have found the dividend. The dividend is x^4 + 3x^3 + 4x^2 + 13x + 12.