Solve a higher-degree polynomial equation or inequality

Higher-degree equations and inequalities

Put the polynomial relation into zero form and factor it completely before finding roots. For an inequality, use the ordered roots to test signs on intervals and apply the strict or inclusive endpoint rule

1. Compare with zero Move all terms to one side before applying the null factor law \(\displaystyle F(x)=G(x)\Rightarrow F(x)-G(x)=0\)
2. Factor completely Keep the scale and every factor so that all real roots can be found \(\displaystyle 2\left(x-1\right)^2\left(x+3\right)=0\)
3. Apply the null factor law If a product is zero, at least one of its factors is zero \(\displaystyle A(x)B(x)=0\Rightarrow A(x)=0\text{ or }B(x)=0\)
4. Use signs and endpoints For an inequality, test each interval and include a root only when equality is allowed \(\displaystyle F(x)\ge0\Rightarrow\text{include the roots where the selected sign interval meets them}\)

Worked example

Solve the polynomial inequality below over the real numbers

1. \(\displaystyle x=-1\text{ with multiplicity }2,\qquad x=3\text{ with multiplicity }1,\qquad x=4\text{ with multiplicity }1\)Set each factor equal to zero and record the exponent at each root
2. \(\displaystyle F(5)=-2\left(6\right)^2\left(2\right)\left(1\right)=-144<0\)Test a value to the right of all three roots to establish the initial sign
3. \(\displaystyle F(x)<0\text{ on }x<-1\text{ and }-1<x<3;\quad F(x)>0\text{ on }3<x<4;\quad F(x)<0\text{ on }x>4\)The even root at −1 does not change the sign, while each simple root changes it
4. \(\displaystyle F(-1)=F(3)=F(4)=0\)The factorised expression is zero at every critical root
5. \(\displaystyle F(x)\ge0\Rightarrow x\in\left\{-1\right\}\cup\left[3,4\right]\)Include the isolated root and the interval from 3 to 4 because the inequality is inclusive

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