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Polynomial Factoring

A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.

Cubics

         

What real value of \(x\) satisfies the equation \[ \frac{x^3-21x^2+20x}{x-1}= -100?\]

The cubic equation \(x^3-7x^2-x+7=0\) has three real roots. Let \(A\) be the greatest root and \(B\) be the least root. What is the value of \(A-B\)?

Given \[2x^3+17x^2+19x-14=(2x-a)(x+b)(x+c),\] what is the value of \(a+b+c\)?

The polynomial in two variables

\[ P(x,y) = x^3 - 10 y^2 + 10 x^2 - y^3 \]

can be expressed as

\[ P(x,y) = (ax+by+c)(dx^2+exy + fy^2 + gx + hy + i ), \]

where the variables from \(a\) to \(i\) are all integers. What is the value of \( | d + e + f + g + h + i | \)?

If \(a,b,\) and \(c\) are positive numbers such that \[x^3-6x^2-37x-30=(x+a)(x+b)(x-c),\] what is the value of \(a+b+c\)?

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