Connect a factorised higher-degree rule and its graph

Factorised higher-degree polynomials

A factorised polynomial displays its roots and multiplicities directly. Use the degree and leading coefficient for the outer ends, then substitute zero to find the vertical intercept

1. Read a root from its factor Set each linear factor equal to zero to find its horizontal-axis intercept \(\displaystyle \left(x-4\right)=0\Rightarrow x=4\)
2. Keep each multiplicity The exponent on a factor is the multiplicity of its root \(\displaystyle \left(x+2\right)^3\Rightarrow x=-2\text{ has multiplicity }3\)
3. Use degree and leading sign Degree parity determines whether the ends agree, and the leading sign determines their direction \(\displaystyle \text{degree }4\text{ and leading coefficient }-2\Rightarrow\text{both ends fall}\)
4. Multiply factor values at zero The vertical intercept is found by substituting zero into every factor and multiplying the values \(\displaystyle P(0)=2\times(0-3)^2\times(0+1)=18\)

Worked example

For the polynomial rule below, state the x-intercepts, their multiplicities and the vertical intercept

1. \(\displaystyle x+1=0\Rightarrow x=-1,\quad x-2=0\Rightarrow x=2,\quad x-3=0\Rightarrow x=3\)Each linear factor is zero at its corresponding x-intercept
2. \(\displaystyle m_{-1}=2,\qquad m_2=1,\qquad m_3=1\)The exponent on each factor is its multiplicity
3. \(\displaystyle \text{degree}=2+1+1=4,\qquad a=-2\Rightarrow\text{both ends fall}\)An even degree gives matching end directions, and a negative leading coefficient sends both ends down
4. \(\displaystyle P(0)=-2\times1^2\times(-2)\times(-3)=-12\)Substitute zero into every factor and multiply the factor values with the leading coefficient

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