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Practice Problems

\(\textbf{1)}\) Write the polynomial equation with roots \(4,\) and \(6\) and leading coefficient \(1\).

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The answer is \( y=x^2-10x+24 \)

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\(\,\,\,\,\,\,\text{when root is }4 \, \Rightarrow \, \text{factor is }(x-4)\)
\(\,\,\,\,\,\,\text{when root is }6 \, \Rightarrow \, \text{factor is }(x-6)\)

\(\,\,\,\,\,\,y=(x-4)(x-6)\)

\(\,\,\,\,\,\,y=x^2-6x-4x+24\)

\(\,\,\,\,\,\,\)The answer is \( y=x^2-10x+24 \)

\(\textbf{2)}\) Write the polynomial equation with roots \(1,\) and \(-3\) and leading coefficient \(1\).

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The answer is \( y=x^2+2x-3 \)

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\(\,\,\,\,\,\,\text{when root is }1 \, \Rightarrow \, \text{factor is }(x-1)\)
\(\,\,\,\,\,\,\text{when root is }-3 \, \Rightarrow \, \text{factor is }(x+3)\)

\(\,\,\,\,\,\,y=(x-1)(x+3)\)

\(\,\,\,\,\,\,y=x^2-1x+3x-3\)

\(\,\,\,\,\,\,\)The answer is \( y=x^2+2x-3 \)

\(\textbf{3)}\) Write the polynomial equation with roots \(0\) and \(2\) and leading coefficient \(1\).

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The answer is \( y=x^2-2x \)

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\(\,\,\,\,\,\,\text{when root is }0 \, \Rightarrow \, \text{factor is }(x)\)
\(\,\,\,\,\,\,\text{when root is }2 \, \Rightarrow \, \text{factor is }(x-2)\)

\(\,\,\,\,\,\,y=(x)(x-2)\)

\(\,\,\,\,\,\,y=x^2-2x\)

\(\,\,\,\,\,\,\)The answer is \( y=x^2-2x \)

\(\textbf{4)}\) Write the polynomial equation with roots \(-3,\) and \(2,\) and \(7\) and leading coefficient \(1\).

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The answer is \( y=x^3+8x^2+x-42 \)

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\(\,\,\,\,\,\,\text{when root is }-3 \, \Rightarrow \, \text{factor is }(x+3)\)
\(\,\,\,\,\,\,\text{when root is }2 \, \Rightarrow \, \text{factor is }(x-2)\)
\(\,\,\,\,\,\,\text{when root is }7 \, \Rightarrow \, \text{factor is }(x-7)\)

\(\,\,\,\,\,\,y=(x+3)(x-2)(x+7)\)

\(\,\,\,\,\,\,y=\left[(x+3)(x-2)\right](x+7)\)

\(\,\,\,\,\,\,y=\left[x^2+3x-2x-6\right](x+7)\)

\(\,\,\,\,\,\,y=\left[x^2+x-6\right](x+7)\)

\(\,\,\,\,\,\,y=x^3+x^2-6x+7x^2+7x-42\)

\(\,\,\,\,\,\,\)The answer is \( y=x^3+8x^2+x-42 \)

\(\textbf{5)}\) Write the polynomial equation with roots \(4,\) and \(3i,\) and leading coefficient \(1\).

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The answer is \( y=x^3-4x^2+9x-36 \)

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\(\,\,\,\,\,\,\text{when root is }4 \, \Rightarrow \, \text{factor is }(x-4)\)
\(\,\,\,\,\,\,\text{when root is }3i \, \Rightarrow \, \text{factor is }(x-3i)\)
\(\,\,\,\,\,\,\text{when root is }-3i \, \Rightarrow \, \text{factor is }(x+3i)\)

\(\,\,\,\,\,\,y=(x-4)(x-3i)(x+3i)\)

\(\,\,\,\,\,\,y=(x-4)(x^2+9)\)

\(\,\,\,\,\,\,y=x^3-4x^2+9x-36\)

\(\,\,\,\,\,\,\)The answer is \( y=x^3-4x^2+9x-36 \)

\(\textbf{6)}\) Write the polynomial equation with roots \(-2,\) and \(5+i,\) and leading coefficient \(1\).

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The answer is \( y=x^3-8x^2+6x+52 \)

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\(\,\,\,\,\,\,\text{when root is }-2 \, \Rightarrow \, \text{factor is }(x+2)\)
\(\,\,\,\,\,\,\text{when root is }5+i \, \Rightarrow \, \text{factor is }(x-5-i)\)
\(\,\,\,\,\,\,\text{when root is }5-i \, \Rightarrow \, \text{factor is }(x-5+i)\)

\(\,\,\,\,\,\,y=(x+2)(x-5-i)(x-5+i)\)

\(\,\,\,\,\,\,y=(x+2)(x^2-10x+26)\)

\(\,\,\,\,\,\,y=x^3-8x^2+6x+52\)

\(\,\,\,\,\,\,\)The answer is \( y=x^3-8x^2+6x+52 \)

See Related Pages\(\)

\(\bullet\text{ Multiply Monomials}\)
\(\,\,\,\,\,\,\,\,(7m^2 k^5 )(8m^3 k^4 )…\)

\(\bullet\text{ Dividing Monomials}\)
\(\,\,\,\,\,\,\,\,\displaystyle \frac{12x^4 y^3 z}{3x^2 z^4 x}…\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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