Question

F(x) = 3x^3 - 2x^2 - 11x + 10; (x + 2)

I need to factor and find the zeros.

Answer 1

`\textsf{Factored function}: \quad f(x)=(x+2)(3x-5)(x-1)`

`\textsf{Zeros}: \quad x=-2, \quad x=(5)/(3),\quad x=1`

Explanation:

Given function:

`f(x)=3x^3-2x^2-11x+10`

If (x + 2) is a factor then:

`\implies f(x)=(x+2)(ax^2+bx+c)`

Expand:

`\implies f(x)=ax^3+bx^2+cx+2ax^2+2bx+2c`

`\implies f(x)=ax^3+2ax^2+bx^2+2bx+cx+2c`

`\implies f(x)=ax^3+(2a+b)x^2+(2b+c)x+2c`

To find a, compare the coefficients of x³:

`\implies a=3`

To find b, substitute the found value of a into the coefficient for x² and compare:

`\implies 2a+b=-2`

`\implies 2(3)+b=-2`

`\implies b=-8`

To find c, compare the constants:

`\implies 2c=10`

`\implies c=5`

Therefore:

`\implies f(x)=(x+2)(3x^2-8x+5)`

Now factor (3x²-8x+5):

`\implies 3x^2-3x-5x+5`

`\implies 3x(x-1)-5(x-1)`

`\implies (3x-5)(x-1)`

Therefore the factored function is:

`\implies f(x)=(x+2)(3x-5)(x-1)`

Zero Product Property

`\boxed{\text{If \; $a \cdot b = 0$ \; then either \; $a = 0$ \;or \;$b = 0$ (or both)}.}`

To find the zeros, set each factor equal to zero and solve for x:

`\implies x+2=0 \implies x=-2`

`\implies 3x-5=0 \implies x=(5)/(3)`

`\implies x-1=0 \implies x=1`

Therefore, the zeros of the function are:

`x=-2, \quad x=(5)/(3),\quad x=1`