Question

Using a given zero to write a polynomial as a product of linear factors: real zeros

For the polynomial below, 3 is a zero.
g(x)=x³ - 3x² - 4x + 12
Express g(x) as a product of linear factors.
g(x) =

Answer 1

`g(x)=(x-3)(x+2)(x-2)`

Explanation:

Given:

• Polynomial: g(x) = x³ - 3x² - 4x + 12
• Zero: 3

Factor Theorem

If f(x) is a polynomial, and f(a) = 0, then (x – a) is a factor of f(x).

Therefore, if 3 is a zero of g(x), then g(3) = 0 and so (x - 3) is a factor of g(x):

`\implies g(x)=(x-3)(ax^2+bx+c)`

As the leading coefficient of g(x) is one, a = 1:

`\implies g(x)=(x-3)(x^2+bx+c)`

As the constant of g(x) is 12, c = 12 ÷ -3 = -4:

`\implies g(x)=(x-3)(x^2+bx-4)`

Expand:

`\implies g(x)=x^3+bx^2-4x-3x^2-3bx+12`

`\implies g(x)=x^3+(b-3)x^2-(4+3b)x+12`

Compare the coefficients of the terms in x² to find b:

`-3x^2=(b-3)x^2 \implies b=0`

Therefore:

`\implies g(x)=(x-3)(x^2-4)`

`\boxed{\begin{minipage}{5 cm}\underline{Difference of Two Squares}\\\\$a^2-b^2=(a+b)(a-b)\\ \end{minipage}}`

To factor (x² - 4), rewrite as (x² - 2²) and apply the difference of two squares:

`\implies g(x)=(x-3)(x+2)(x-2)`

Therefore, the function g(x) as a product of linear factors is:

• 
  `g(x)=(x-3)(x+2)(x-2)`

Check by expanding the factored function:

`\implies g(x)=(x-3)(x+2)(x-2)`

`\implies g(x)=(x^2-x-6)(x-2)`

`\implies g(x)=x^3-2x^2-x^2+2x-6x+12`

`\implies g(x)=x^3-3x^2-4x+12`