To express [tex]\( g(x) = x^3 + 7x^2 + 20x + 24 \)[/tex] as a product of linear factors given that [tex]\(-3\)[/tex] is a zero, follow these steps:
1. Verify that [tex]\(-3\)[/tex] is a zero:
Substitute [tex]\( x = -3 \)[/tex] into the polynomial to see if it equals zero.
[tex]\[
g(-3) = (-3)^3 + 7(-3)^2 + 20(-3) + 24
\][/tex]
Calculate each term:
[tex]\[
(-3)^3 = -27
\][/tex]
[tex]\[
7(-3)^2 = 7(9) = 63
\][/tex]
[tex]\[
20(-3) = -60
\][/tex]
Summing these values, we get:
[tex]\[
g(-3) = -27 + 63 - 60 + 24 = 0
\][/tex]
Hence, [tex]\(-3\)[/tex] is indeed a zero of the polynomial.
2. Perform polynomial division of [tex]\( g(x) \)[/tex] by [tex]\( (x + 3) \)[/tex]:
To factorize [tex]\( g(x) \)[/tex] by [tex]\( (x + 3) \)[/tex], we use synthetic division or long division. We'll use synthetic division here.
[tex]\(-3\)[/tex] | [tex]\( 1 \quad 7 \quad 20 \quad 24 \)[/tex]
Perform synthetic division:
- Bring down the leading coefficient [tex]\(1\)[/tex].
- Multiply [tex]\(-3\)[/tex] by the current coefficient in the division (start with [tex]\(1\)[/tex]) and add it to the next coefficient.
- Repeat the step for all coefficients.
[tex]\[
\begin{array}{r|rrrr}
-3 & 1 & 7 & 20 & 24 \\
\hline
& & -3 & -12 & -24 \\
\hline
& 1 & 4 & 8 & 0 \\
\end{array}
\][/tex]
The coefficients of the quotient polynomial are [tex]\(1, 4, 8\)[/tex], which corresponds to [tex]\( x^2 + 4x + 8 \)[/tex].
Therefore,
[tex]\[
\frac{x^3 + 7x^2 + 20x + 24}{x + 3} = x^2 + 4x + 8
\][/tex]
So,
[tex]\[
g(x) = (x + 3)(x^2 + 4x + 8)
\][/tex]
3. Factorize the quadratic polynomial [tex]\( x^2 + 4x + 8 \)[/tex]:
To factorize [tex]\(x^2 + 4x + 8\)[/tex], we look for factors that satisfy the form [tex]\((x - r_1)(x - r_2)\)[/tex], where [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex] are the roots of the quadratic equation.
The quadratic equation [tex]\(x^2 + 4x + 8 = 0\)[/tex] can be solved using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 1, b = 4, \)[/tex] and [tex]\( c = 8 \)[/tex].
[tex]\[
x = \frac{-4 \pm \sqrt{4^2 - 4(1)(8)}}{2(1)} = \frac{-4 \pm \sqrt{16 - 32}}{2} = \frac{-4 \pm \sqrt{-16}}{2} = \frac{-4 \pm 4i}{2} = -2 \pm 2i
\][/tex]
Thus, the roots of [tex]\(x^2 + 4x + 8\)[/tex] are [tex]\( -2 + 2i \)[/tex] and [tex]\( -2 - 2i \)[/tex].
Therefore:
[tex]\[
x^2 + 4x + 8 = (x - (-2 + 2i))(x - (-2 - 2i)) = (x + 2 - 2i)(x + 2 + 2i)
\][/tex]
4. Combine all factors:
Finally, we have:
[tex]\[
g(x) = (x + 3)(x + 2 - 2i)(x + 2 + 2i)
\][/tex]
Therefore, [tex]\( g(x) = x^3 + 7x^2 + 20x + 24 \)[/tex] expressed as a product of linear factors is:
[tex]\[
g(x) = (x + 3)(x + 2 - 2i)(x + 2 + 2i)
\][/tex]
