Solving Cubic Equations

Express the cubic equation as \(f(x)=0\). Use trial and error to find a root. Alternatively, use a root given in an earlier part. If \(f(a)=0\), then \((x-a)\) is a factor.

Solve \(4x^3+2=5x^2+7x\).

\(4x^3-5x^2-7x+2=0\)

Let \(f(x)=4x^3-5x^2-7x+2\).

\(f(-1)=4(-1)^3-5(-1)^2-7(-1)+2=0\)

Therefore \(x=-1\) is a root and \((x+1)\) is a factor.

Divide the cubic expression by the linear factor. Alternatively, write the cubic as \((x-a)(Ax^2+Bx+C)\) and compare coefficients.

Dividing \(4x^3-5x^2-7x+2\) by \((x+1)\) gives \(4x^2-9x+2\).

Hence \(4x^3-5x^2-7x+2=(x+1)(4x^2-9x+2)\).

Factorise the quadratic expression. If it cannot be factorised readily, use the quadratic formula.

\((x+1)(4x^2-9x+2)=0\)

\((x+1)(4x-1)(x-2)=0\)

Therefore \(x=-1\), \(x=\frac{1}{4}\) or \(x=2\).