[tex]f(x)=(2x+3)(x-1)(x-4)[/tex]
a) To find the root we have to set the function equal to zero
[tex]\begin{gathered} (2x+3)(x-1)(x-4)=0 \\ \end{gathered}[/tex][tex]\begin{gathered} (2x+3)=0 \\ x_1=-\frac{3}{2} \end{gathered}[/tex][tex]\begin{gathered} x-1=0 \\ x_2=1 \end{gathered}[/tex][tex]\begin{gathered} x-4=0 \\ x_3=4 \end{gathered}[/tex]
b) The intervals obtained when the x-intercepts are used to partition the number line are
[tex]\begin{gathered} (-\infty,-\frac{3}{2})\to1 \\ (-\frac{3}{2},1)\to2 \\ (1,4)\to3 \\ (4,+\infty)\to4 \end{gathered}[/tex]
c) The table of signs
d) A sketch of the graph
What happens to the graph as x decreases?
• The graph tends to negative infinity
What happens to the graph as x increments?
• The graph tends to negative infinity
For which intervals is the graph above of x axis
[tex]\begin{gathered} (-\frac{3}{2},1) \\ (4,+\infty) \end{gathered}[/tex]
For which intervals is the graph below of x axis
[tex]\begin{gathered} (-\infty,-\frac{3}{2}) \\ (1,4) \end{gathered}[/tex]
What is the leading term of the polynomial function?
To calculate the leading term we must expand the function
[tex](2x+3)(x-1)(x-4)=2x^3-7x^2-7x+12[/tex]
The leading term is the term with the highest exponent in this case x³
What is the leading coefficient of the polynomial function?
It is the number that accompanies the leading term in this case 2
