Using the factors (-5x+4) and (x+1), find the remaining factor (s) of f(x)=-20x^4+91x^3+60x^2-71x-20 and write the polynomial in full factored form.
To find the remaining factor(s) of the polynomial f(x) = -20x^4 + 91x^3 + 60x^2 - 71x - 20, we can perform polynomial long division or synthetic division with the factors we already have, (-5x + 4) and (x + 1).
Let's start with the factor (-5x + 4):
1. Set up the long division or synthetic division as follows:
-5x + 4 | -20x^4 + 91x^3 + 60x^2 - 71x - 20
2. Divide the first term of the polynomial by the first term of the factor, -20x^4 / -5x = 4x^3, and write this result above the line.
-5x + 4 | -20x^4 + 91x^3 + 60x^2 - 71x - 20
4x^3
3. Multiply the divisor (-5x + 4) by the result, 4x^3, and write the product underneath the polynomial, aligning the like terms properly:
-5x + 4 | -20x^4 + 91x^3 + 60x^2 - 71x - 20
4x^3
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-20x^4 + 16x^3
4. Subtract the product from the original polynomial:
-5x + 4 | -20x^4 + 91x^3 + 60x^2 - 71x - 20
4x^3
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-20x^4 + 16x^3
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75x^3 + 60x^2 - 71x - 20
5. Repeat steps 2-4 with the new polynomial, 75x^3 + 60x^2 - 71x - 20, and the remaining factor (x + 1).
x + 1 | 75x^3 + 60x^2 - 71x - 20
?
Performing the division, we find that the quotient is 75x^2 - 15x + 4.
Therefore, the full factored form of the polynomial f(x) = -20x^4 + 91x^3 + 60x^2 - 71x - 20 is:
f(x) = (-5x + 4)(x + 1)(4x^3 + 75x^2 - 15x + 4)