Suppose that a polynomial function of degree 4

${\displaystyle -1}$ , ${\displaystyle \sqrt{3}}$ , ${\displaystyle \frac{11}{3}}$

Roots are the points where the graph intercepts with the x-axis ${\displaystyle (y=0)}$.

${\displaystyle y=0}$ at the roots

The root at ${\displaystyle x=-1}$ was found by solving for ${\displaystyle x}$ when ${\displaystyle x-(-1)=y}$ and ${\displaystyle y=0}$.

The factor is ${\displaystyle x+1}$

The root at ${\displaystyle x=\sqrt{3}}$ was found by solving for ${\displaystyle x}$ when ${\displaystyle x-\left(\sqrt{3}\right)=y}$ and ${\displaystyle y=0}$.

The factor is ${\displaystyle x-\sqrt{3}}$

The root at ${\displaystyle x=\frac{11}{3}}$ was found by solving for ${\displaystyle x}$ when ${\displaystyle x-\left(\frac{11}{3}\right)=y}$ and ${\displaystyle y=0}$.

The factor is ${\displaystyle x-\frac{11}{3}}$

Combine all the factors into a single equation.

${\displaystyle y=(x+1)(x-\sqrt{3})(x-\frac{11}{3})}$

Multiply all the factors to simplify the equation ${\displaystyle y=(x+1)(x-\sqrt{3})(x-\frac{11}{3})}$.

Expand ${\displaystyle (x+1)(x-\sqrt{3})}$ using the FOIL Method.

Apply the distributive property.

${\displaystyle y=(x(x-\sqrt{3})+1(x-\sqrt{3}))(x-\frac{11}{3})}$

Apply the distributive property.

${\displaystyle y=(x\cdot x+x(-\sqrt{3})+1(x-\sqrt{3}))(x-\frac{11}{3})}$

Apply the distributive property.

${\displaystyle y=(x\cdot x+x(-\sqrt{3})+1x+1(-\sqrt{3}))(x-\frac{11}{3})}$

Simplify each term.

Multiply ${\displaystyle x}$ by ${\displaystyle x}$.

${\displaystyle y=(x^2+x(-\sqrt{3})+1x+1(-\sqrt{3}))(x-\frac{11}{3})}$

Multiply x${\displaystyle x}$ by 1${\displaystyle 1}$.

${\displaystyle y=(x^2+x(-\sqrt{3})+x+1(-\sqrt{3}))(x-\frac{11}{3})}$

Multiply ${\displaystyle -\sqrt{3}}$ by ${\displaystyle 1}$.

${\displaystyle y=(x^2+x(-\sqrt{3})+x-\sqrt{3})(x-\frac{11}{3})}$

Expand ${\displaystyle (x^2+x(-\sqrt{3})+x-\sqrt{3})(x-\frac{11}{3})}$ by multiplying each term in the first expression by each term in the second expression.

${\displaystyle y=x^{2}x+x^2(-\frac{11}{3})+x(-\sqrt{3})x+x(-\sqrt{3})(-\frac{11}{3})+x\cdot x+x(-\frac{11}{3})-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Simplify each term.

Multiply ${\displaystyle x^2}$ by ${\displaystyle x}$ by adding the exponents.

Tap for more steps...

${\displaystyle y=x^3+x^2(-\frac{11}{3})+x(-\sqrt{3})x+x(-\sqrt{3})(-\frac{11}{3})+x\cdot x+x(-\frac{11}{3})-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Combine ${\displaystyle x^2}$ and ${\displaystyle \frac{11}{3}}$.

${\displaystyle y=x^3-\frac{x^2\cdot 11}{3}+x(-\sqrt{3})x+x(-\sqrt{3})(-\frac{11}{3})+x\cdot x+x(-\frac{11}{3})-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Move ${\displaystyle 11}$ to the left of ${\displaystyle x^2}$.

${\displaystyle y=x^3-\frac{11\cdot x^2}{3}+x(-\sqrt{3})x+x(-\sqrt{3})(-\frac{11}{3})+x\cdot x+x(-\frac{11}{3})-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Multiply ${\displaystyle x}$ by ${\displaystyle x}$ by adding the exponents.

${\displaystyle y=x^3-\frac{11x^2}{3}+x^2(-\sqrt{3})+x(-\sqrt{3})(-\frac{11}{3})+x\cdot x+x(-\frac{11}{3})-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Multiply ${\displaystyle x(-\sqrt{3})(-\frac{11}{3})}$.

${\displaystyle y=x^3-\frac{11x^2}{3}+x^2(-\sqrt{3})+\frac{\sqrt{3}(x\cdot 11)}{3}+x\cdot x+x(-\frac{11}{3})-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Move ${\displaystyle 11}$ to the left of ${\displaystyle \sqrt{3}x}$.

${\displaystyle y=x^3-\frac{11x^2}{3}+x^2(-\sqrt{3})+\frac{11\cdot \left(\sqrt{3}x\right)}{3}+x\cdot x+x(-\frac{11}{3})-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Multiply ${\displaystyle x}$ by ${\displaystyle x}$.

${\displaystyle y=x^3-\frac{11x^2}{3}+x^2(-\sqrt{3})+\frac{11\sqrt{3}x}{3}+x^2+x(-\frac{11}{3})-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Combine ${\displaystyle x}$ and ${\displaystyle \frac{11}{3}}$.

${\displaystyle y=x^3-\frac{11x^2}{3}+x^2(-\sqrt{3})+\frac{11\sqrt{3}x}{3}+x^2-\frac{x\cdot 11}{3}-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Move ${\displaystyle 11}$ to the left of ${\displaystyle x}$.

${\displaystyle y=x^3-\frac{11x^2}{3}+x^2(-\sqrt{3})+\frac{11\sqrt{3}x}{3}+x^2-\frac{11\cdot x}{3}-\sqrt{3}x-\sqrt{3}(-\frac{11}{3})}$

Multiply${\displaystyle -\sqrt{3}(-\frac{11}{3})}$.

${\displaystyle y=x^3-\frac{11x^2}{3}+x^2(-\sqrt{3})+\frac{11\sqrt{3}x}{3}+x^2-\frac{11x}{3}-\sqrt{3}x+\frac{\sqrt{3}\cdot 11}{3}}$

Move ${\displaystyle 11}$ to the left of ${\displaystyle \sqrt{3}}$.

${\displaystyle y=x^3-\frac{11x^2}{3}+x^2(-\sqrt{3})+\frac{11\sqrt{3}x}{3}+x^2-\frac{11x}{3}-\sqrt{3}x+\frac{11\sqrt{3}}{3}}$

To write ${\displaystyle x^2}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{3}{3}}$.

${\displaystyle y=x^3+x^2(-\sqrt{3})+\frac{11\sqrt{3}x}{3}-\frac{11x^2}{3}+x^2\cdot \frac{3}{3}-\frac{11x}{3}-\sqrt{3}x+\frac{11\sqrt{3}}{3}}$

Simplify terms.

Combine ${\displaystyle x^2}$ and ${\displaystyle \frac{3}{3}}$.

${\displaystyle y=x^3+x^2(-\sqrt{3})+\frac{11\sqrt{3}x}{3}-\frac{11x^2}{3}+\frac{x^2\cdot 3}{3}-\frac{11x}{3}-\sqrt{3}x+\frac{11\sqrt{3}}{3}}$

Combine the numerators over the common denominator.

${\displaystyle y=x^3+x^2(-\sqrt{3})+\frac{11\sqrt{3}x}{3}+\frac{-11x^2+x^2\cdot 3}{3}-\frac{11x}{3}-\sqrt{3}x+\frac{11\sqrt{3}}{3}}$

Combine the numerators over the common denominator.

${\displaystyle y=x^3+x^2(-\sqrt{3})-\sqrt{3}x+\frac{11\sqrt{3}x-11x^2+x^2\cdot 3-11x+11\sqrt{3}}{3}}$

Move ${\displaystyle 3}$ to the left of ${\displaystyle x^2}$.

${\displaystyle y=x^3+x^2(-\sqrt{3})-\sqrt{3}x+\frac{11\sqrt{3}x-11x^2+3x^2-11x+11\sqrt{3}}{3}}$

Simplify by adding terms.

Add ${\displaystyle -11x^2}$ and ${\displaystyle 3x^2}$.

${\displaystyle y=x^3+x^2(-\sqrt{3})-\sqrt{3}x+\frac{11\sqrt{3}x-8x^2-11x+11\sqrt{3}}{3}}$

Reorder ${\displaystyle 11\sqrt{3}x}$ and ${\displaystyle -8x^2}$.

${\displaystyle y=x^3+x^2(-\sqrt{3})-\sqrt{3}x+\frac{-8x^2+11\sqrt{3}x-11x+11\sqrt{3}}{3}}$

To write ${\displaystyle x^3}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{3}{3}}$.

${\displaystyle y=x^2(-\sqrt{3})-\sqrt{3}x+x^3\cdot \frac{3}{3}+\frac{-8x^2+11\sqrt{3}x-11x+11\sqrt{3}}{3}}$

Simplify terms.

Combine ${\displaystyle x^3}$ and ${\displaystyle \frac{3}{3}}$.

${\displaystyle y=x^2(-\sqrt{3})-\sqrt{3}x+\frac{x^3\cdot 3}{3}+\frac{-8x^2+11\sqrt{3}x-11x+11\sqrt{3}}{3}}$

Combine the numerators over the common denominator.

${\displaystyle y=x^2(-\sqrt{3})-\sqrt{3}x+\frac{x^3\cdot 3-8x^2+11\sqrt{3}x-11x+11\sqrt{3}}{3}}$

Move ${\displaystyle 3}$ to the left of ${\displaystyle x^3}$.

${\displaystyle y=x^2(-\sqrt{3})-\sqrt{3}x+\frac{3x^3-8x^2+11\sqrt{3}x-11x+11\sqrt{3}}{3}}$

To write ${\displaystyle x^2(-\sqrt{3})}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{3}{3}}$.

${\displaystyle y=-\sqrt{3}x+x^2(-\sqrt{3})\cdot \frac{3}{3}+\frac{3x^3-8x^2+11\sqrt{3}x-11x+11\sqrt{3}}{3}}$

Simplify terms.

Combine ${\displaystyle x^2(-\sqrt{3})}$ and ${\displaystyle \frac{3}{3}}$.

${\displaystyle y=-\sqrt{3}x+\frac{x^2(-\sqrt{3})\cdot 3}{3}+\frac{3x^3-8x^2+11\sqrt{3}x-11x+11\sqrt{3}}{3}}$

Combine the numerators over the common denominator.

${\displaystyle y=-\sqrt{3}x+\frac{x^2(-\sqrt{3})\cdot 3+3x^3-8x^2+11\sqrt{3}x-11x+11\sqrt{3}}{3}}$

Simplify the numerator.

Multiply ${\displaystyle 3}$ by ${\displaystyle -1}$.

${\displaystyle y=-\sqrt{3}x+\frac{x^2(-3\sqrt{3})+3x^3-8x^2+11\sqrt{3}x-11x+11\sqrt{3}}{3}}$

Reorder terms.

${\displaystyle y=-\sqrt{3}x+\frac{3x^3-8x^2-3x^2\sqrt{3}+11x\sqrt{3}-11x+11\sqrt{3}}{3}}$

To write ${\displaystyle -\sqrt{3}x}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{3}{3}}$.${\displaystyle y=-\sqrt{3}x\cdot \frac{3}{3}+\frac{3x^3-8x^2-3x^2\sqrt{3}+11x\sqrt{3}-11x+11\sqrt{3}}{3}}$

Simplify terms.

Combine ${\displaystyle -\sqrt{3}x}$ and ${\displaystyle \frac{3}{3}}$.

${\displaystyle y=\frac{-\sqrt{3}x\cdot 3}{3}+\frac{3x^3-8x^2-3x^2\sqrt{3}+11x\sqrt{3}-11x+11\sqrt{3}}{3}}$

Combine the numerators over the common denominator.

${\displaystyle y=\frac{-\sqrt{3}x\cdot 3+3x^3-8x^2-3x^2\sqrt{3}+11x\sqrt{3}-11x+11\sqrt{3}}{3}}$

Simplify the numerator.

Multiply ${\displaystyle 3}$ by ${\displaystyle -1}$.

${\displaystyle y=\frac{-3\sqrt{3}x+3x^3-8x^2-3x^2\sqrt{3}+11x\sqrt{3}-11x+11\sqrt{3}}{3}}$

Reorder the factors of ${\displaystyle -3\sqrt{3}x}$.

${\displaystyle y=\frac{-3x\sqrt{3}+3x^3-8x^2-3x^2\sqrt{3}+11x\sqrt{3}-11x+11\sqrt{3}}{3}}$

Add ${\displaystyle -3x\sqrt{3}}$ and ${\displaystyle 11x\sqrt{3}}$.

${\displaystyle y=\frac{3x^3-8x^2-3x^2\sqrt{3}+8x\sqrt{3}-11x+11\sqrt{3}}{3}}$

Split the fraction ${\displaystyle \frac{3x^3-8x^2-3x^2\sqrt{3}+8x\sqrt{3}-11x+11\sqrt{3}}{3}}$ into two fractions.

${\displaystyle y=\frac{3x^3-8x^2-3x^2\sqrt{3}+8x\sqrt{3}-11x}{3}+\frac{11\sqrt{3}}{3}}$

Split the fraction ${\displaystyle \frac{3x^3-8x^2-3x^2\sqrt{3}+8x\sqrt{3}-11x}{3}}$ into two fractions.

${\displaystyle y=\frac{3x^3-8x^2-3x^2\sqrt{3}+8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Split the fraction ${\displaystyle \frac{3x^3-8x^2-3x^2\sqrt{3}+8x\sqrt{3}}{3}}$ into two fractions.

${\displaystyle y=\frac{3x^3-8x^2-3x^2\sqrt{3}}{3}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Split the fraction ${\displaystyle \frac{3x^3-8x^2-3x^2\sqrt{3}}{3}}$ into two fractions.

${\displaystyle y=\frac{3x^3-8x^2}{3}+\frac{-3x^2\sqrt{3}}{3}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Split the fraction ${\displaystyle \frac{3x^3-8x^2}{3}}$ into two fractions.

${\displaystyle y=\frac{3x^3}{3}+\frac{-8x^2}{3}+\frac{-3x^2\sqrt{3}}{3}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Cancel the common factor of ${\displaystyle 3}$.

${\displaystyle y=\frac{\cancel{3}{x}^3}{\cancel{3}}+\frac{-8x^2}{3}+\frac{-3x^2\sqrt{3}}{3}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Divide ${\displaystyle x^3}$ by ${\displaystyle 1}$.

${\displaystyle y=x^3+\frac{-8x^2}{3}+\frac{-3x^2\sqrt{3}}{3}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Move the negative in front of the fraction.

${\displaystyle y=x^3-\frac{8x^2}{3}+\frac{-3x^2\sqrt{3}}{3}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Cancel the common factor of ${\displaystyle -3}$ and ${\displaystyle 3}$.

Factor ${\displaystyle 3}$ out of ${\displaystyle -3x^2\sqrt{3}}$.

${\displaystyle y=x^3-\frac{8x^2}{3}+\frac{3(-x^2\sqrt{3})}{3}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Cancel the common factors.

Factor ${\displaystyle 3}$ out of ${\displaystyle 3}$.

${\displaystyle y=x^3-\frac{8x^2}{3}+\frac{3(-x^2\sqrt{3})}{3\left(1\right)}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Cancel the common factor.

${\displaystyle y=x^3-\frac{8x^2}{3}+\frac{\cancel{3}(-x^2\sqrt{3})}{\cancel{3}\cdot 1}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Rewrite the expression.

${\displaystyle y=x^3-\frac{8x^2}{3}+\frac{-x^2\sqrt{3}}{1}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Divide ${\displaystyle -x^2\sqrt{3}}$ by ${\displaystyle 1}$.

${\displaystyle y=x^3-\frac{8x^2}{3}-x^2\sqrt{3}+\frac{8x\sqrt{3}}{3}+\frac{-11x}{3}+\frac{11\sqrt{3}}{3}}$

Move the negative in front of the fraction.

${\displaystyle y=x^3-\frac{8x^2}{3}-x^2\sqrt{3}+\frac{8x\sqrt{3}}{3}-\frac{11x}{3}+\frac{11\sqrt{3}}{3}}$