Certainly! Let's solve the problem step by step to find the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
We start with the given polynomial:
[tex]\[ 2x^3 - 2x^2 - 24x \][/tex]
We need to express it in the form:
[tex]\[ ax(x+b)(x+c) \][/tex]
First, let's expand [tex]\( ax(x+b)(x+c) \)[/tex]:
[tex]\[ ax(x+b)(x+c) = ax(x^2 + (b+c)x + bc) \][/tex]
Distribute [tex]\( ax \)[/tex] through the expression inside the parentheses:
[tex]\[ = ax^3 + a(b+c)x^2 + abc \][/tex]
Now we compare coefficients of the expanded polynomial [tex]\( ax^3 + a(b+c)x^2 + abc \)[/tex] with the given polynomial [tex]\( 2x^3 - 2x^2 -24x \)[/tex].
By comparing coefficients, we get the following system of equations:
1. The coefficient of [tex]\( x^3 \)[/tex]: [tex]\( a = 2 \)[/tex]
2. The coefficient of [tex]\( x^2 \)[/tex]: [tex]\( a(b + c) = -2 \)[/tex]
3. The coefficient of [tex]\( x \)[/tex]: [tex]\( abc = -24 \)[/tex]
From the first equation, we know:
[tex]\[ a = 2 \][/tex]
Now substitute [tex]\( a = 2 \)[/tex] into the second and third equations:
[tex]\[ 2(b + c) = -2 \][/tex]
[tex]\[ 2bc = -24 \][/tex]
Simplifying these equations gives us:
[tex]\[ b + c = -1 \][/tex]
[tex]\[ bc = -12 \][/tex]
Now, we need to solve this system of equations for [tex]\( b \)[/tex] and [tex]\( c \)[/tex].
To find the values of [tex]\( b \)[/tex] and [tex]\( c \)[/tex], we'll solve the quadratic equation derived from [tex]\( b + c = -1 \)[/tex] and [tex]\( bc = -12 \)[/tex]:
[tex]\[ t^2 + (b + c)t + bc = 0 \][/tex]
[tex]\[ t^2 - t - 12 = 0 \][/tex]
Now, solve the quadratic equation [tex]\( t^2 - t - 12 = 0 \)[/tex]:
[tex]\[
t = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1}
\][/tex]
[tex]\[
t = \frac{1 \pm \sqrt{1 + 48}}{2}
\][/tex]
[tex]\[
t = \frac{1 \pm \sqrt{49}}{2}
\][/tex]
[tex]\[
t = \frac{1 \pm 7}{2}
\][/tex]
This yields two solutions:
[tex]\[
t = \frac{1 + 7}{2} = 4
\][/tex]
[tex]\[
t = \frac{1 - 7}{2} = -3
\][/tex]
Therefore, the values of [tex]\( b \)[/tex] and [tex]\( c \)[/tex] are:
[tex]\[ b = 4, \; c = -3 \][/tex]
or
[tex]\[ b = -3, \; c = 4 \][/tex]
So, the missing values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = 4, \; c = -3 \][/tex]
or
[tex]\[ b = -3, \; c = 4 \][/tex]
In summary:
[tex]\[
a = 2, \; (b, c) = (4, -3) \; \text{or} \; (-3, 4)
\][/tex]
