Notes

Problems

For each function, find the roots and their corresponding multiplicity.

\(\textbf{1)}\) \( f(x)=x^2+5x-6 \)

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Factored: \( (x+6)(x-1) \)
Roots: \(-6 \) (Multiplicity 1), \(1 \) (Multiplicity 1)

 

\(\textbf{2)}\) \( f(x)=x^3-5x^2+6x \)

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Factored: \( x(x-2)(x-3) \)
Roots: \(0 \) (Multiplicity 1), \(2 \) (Multiplicity 1), \(3 \) (Multiplicity 1)

 

\(\textbf{3)}\) \( f(x)= x^5-2x^4-4x^3+8x^2 \)

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Factored: \( x^2(x-2)^2(x+2) \)
Roots: \(-2 \) (Multiplicity 1), \(0 \) (Multiplicity 2), \(2 \) (Multiplicity 2)

 

\(\textbf{4)}\) \( f(x)=x^2-6x+9 \)

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Factored: \( (x-3)^2 \)
Roots: \(3 \) (Multiplicity 2)

 

\(\textbf{5)}\) \( f(x)= 3x(x-3)^2(x+1)^4 \)

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Factored: \( 3x(x-3)^2(x+1)^4 \)
Roots: \(-1 \) (Multiplicity 4), \(0 \) (Multiplicity 1), \(3 \) (Multiplicity 2)

 

\(\textbf{6)}\) \( f(x)=x^3(x+3)(x+1)^2 \)

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Factored: \( x^3(x+3)(x+1)^2 \)
Roots: \(-3 \) (Multiplicity 1), \(-1 \) (Multiplicity 2), \(0 \) (Multiplicity 3)

 

\(\textbf{7)}\) \( f(x)=x^2-1 \)

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Factored: \( (x+1)(x-1) \)
Roots: \(-1 \) (Multiplicity 1), \(1 \) (Multiplicity 1)

 

See Related Pages\(\)

\(\bullet\text{ Adding and Subtracting Polynomials}\)
\(\,\,\,\,\,\,\,\,(4d+7)−(2d−5)…\)

\(\bullet\text{ Multiplying Polynomials}\)
\(\,\,\,\,\,\,\,\,(x+2)(x^2+3x−5)…\)

\(\bullet\text{ Dividing Polynomials}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)

\(\bullet\text{ Dividing Polynomials (Synthetic Division)}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)

\(\bullet\text{ Synthetic Substitution}\)
\(\,\,\,\,\,\,\,\,f(x)=4x^4−3x^2+8x−2…\)

\(\bullet\text{ End Behavior}\)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow \infty, \quad f(x)\rightarrow \infty \)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow -\infty, \quad f(x)\rightarrow \infty… \)

\(\bullet\text{ Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+10x−24=0…\)

\(\bullet\text{ Quadratic Formula and the Discriminant}\)
\(\,\,\,\,\,\,\,\,x=-b \pm \displaystyle\frac{\sqrt{b^2-4ac}}{2a}…\)

\(\bullet\text{ Complex Numbers}\)
\(\,\,\,\,\,\,\,\,i=\sqrt{-1}…\)

\(\bullet\text{ Multiplicity of Roots}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Rational Zero Theorem}\)
\(\,\,\,\,\,\,\,\, \pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12…\)

\(\bullet\text{ Descartes Rule of Signs}\)
\(\,\)

\(\bullet\text{ Roots and Zeroes}\)
\(\,\,\,\,\,\,\,\,\text{Solve for }x. 3x^2+4x=0…\)

\(\bullet\text{ Linear Factored Form}\)
\(\,\,\,\,\,\,\,\,f(x)=(x+4)(x+1)(x−3)…\)

\(\bullet\text{ Polynomial Inequalities}\)
\(\,\,\,\,\,\,\,\,x^3-4x^2-4x+16 \gt 0…\)