Given the polynomial:
[tex]g(x)=x^3+9x+20x+6[/tex]
Where:
-3 is a zero of the polynomial.
Let's express the function as a product of linear factors.
Where:
x = -3
Equate to zero:
x + 3 = 0
Using synthetic division, let's divide the polynomial by -3:
Therefore, we have:
[tex]g(x)=(x+3)(x^2+6x+2)[/tex]
Now,let's solve the expression(quotient) for x using the quadratic formula
[tex]x^2+6x+2[/tex]
Apply the general quadratic equation:
[tex]\begin{gathered} ax^2+bx+c \\ \\ x^2+6x+2 \end{gathered}[/tex]
WHere:
a = 1
b = 6
c = 2
Apply the quadratic formula:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]
Hence, we have:
[tex]\begin{gathered} x=\frac{-6\pm\sqrt[]{6^2-4(1)(2)}}{2(1)} \\ \\ x=\frac{-6\pm\sqrt[]{36-8}}{2} \\ \\ x=\frac{-6\pm\sqrt[]{28}}{2} \\ \\ x=\frac{-6}{2}\pm\frac{\sqrt[]{28}}{2} \\ \\ x=-3\pm\frac{\sqrt[]{4\cdot7}}{2} \\ \\ x=-3\pm\frac{2\sqrt[]{7}}{2} \\ \\ x=-3\pm\sqrt[]{7} \\ \\ x=-3-\sqrt[]{7},-3+\sqrt[]{7} \end{gathered}[/tex]
Therefore, the polynomial g(x) as a product of linear factors are:
[tex]g(x)=(x+3)(x+3-\sqrt[]{7})(x+3+\sqrt[]{7})[/tex]
ANSWER:
[tex]g(x)=(x+3)(x+3-\sqrt[]{7})(x+3+\sqrt[]{7})[/tex]
