A polynomial is an algebraic expression involving many terms and can be factorised using long division or synthetic division.
Part ofMathsAlgebraic and trigonometric skills
Factorise fully \(2{x^4} + 9{x^3} - 18{x^2} - 71x - 30\)
Let's try \((x \pm 1)\) first, that is \(x = \pm 1\).
\(f(x) = 2{x^4} + 9{x^3} - 18{x^2} - 71x - 30\)
\(f(1) = 2{(1)^4} + 9{(1)^3} - 18{(1)^2} - 71(1) - 30\)
\(= - 108\)
\(\ne 0\)
\(\Rightarrow (x - 1)\) is not a factor
\(f( - 1) = 2{( - 1)^4} + 9{( - 1)^3} - 18{( - 1)^2} - 71( - 1) - 30\)
\(= 16\)
\(\Rightarrow (x + 1)\) is not a factor
Next we will try \((x \pm 2)\), or \(x = \pm 2\)
\(f(2) = 2{(2)^4} + 9{(2)^3} - 18{(2)^2} - 71(2) - 30\)
\(= - 140\)
\(\Rightarrow (x - 2)\) is not a factor
\(f( - 2) = 2{( - 2)^4} + 9{( - 2)^3} - 18{( - 2)^2} - 71( - 2) - 30\)
\(= 0\)
\(\Rightarrow (x + 2)\) is a factor
