To write the polynomial [tex]\( g(x) = x^2 + 10x + 18 \)[/tex] as the product of linear factors and find all the zeros of the function, follow these steps:
### Step 1: Factor the polynomial
1. Identify the polynomial:
[tex]\[ g(x) = x^2 + 10x + 18 \][/tex]
2. Find two numbers that multiply to the constant term (18) and add up to the coefficient of the linear term (10).
We need to find two numbers, let's call them [tex]\( a \)[/tex] and [tex]\( b \)[/tex], such that:
[tex]\[ a \times b = 18 \][/tex]
[tex]\[ a + b = 10 \][/tex]
After some trial and error, we find that the numbers 2 and 9 work:
[tex]\[ 2 \times 9 = 18 \][/tex]
[tex]\[ 2 + 9 = 10 \][/tex]
3. Rewrite the middle term using the identified numbers:
[tex]\[ x^2 + 10x + 18 = x^2 + 2x + 9x + 18 \][/tex]
4. Factor by grouping:
[tex]\[ x^2 + 2x + 9x + 18 = x(x + 2) + 9(x + 2) \][/tex]
5. Factor out the common binomial factor:
[tex]\[ x(x + 2) + 9(x + 2) = (x + 9)(x + 2) \][/tex]
Therefore, the polynomial [tex]\( g(x) \)[/tex] expressed as the product of linear factors is:
[tex]\[ g(x) = (x + 9)(x + 2) \][/tex]
### Step 2: Find the zeros of the function
To find the zeros (roots) of the function, set [tex]\( g(x) \)[/tex] equal to zero:
[tex]\[ (x + 9)(x + 2) = 0 \][/tex]
Solve for [tex]\( x \)[/tex] by setting each factor equal to zero:
[tex]\[ x + 9 = 0 \][/tex]
[tex]\[ x = -9 \][/tex]
[tex]\[ x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
### Answer
The polynomial [tex]\( g(x) = x^2 + 10x + 18 \)[/tex] can be factored as:
[tex]\[ g(x) = (x + 9)(x + 2) \][/tex]
The zeros of the function are:
[tex]\[ x = -9, -2 \][/tex]
