The polynomial
\[ \begin{align}
p(x) &= x^3 + ax^2 + bx - 4 \quad
\end{align} \ \]
has \( (x^2 - 4) \) as a factor. Find the values of a and b and then factor p(x).
Re-express p(x) using the known factors.
\[ \begin{align}
x^3 + ax^2 + bx - 4 &= (x^2 - 4)q(x) \\
&= (x - 2)(x + 2)q(x) \\
\end{align} \ \]
So 2 and -2 are both factors of p(x).
Find two equations in the unknowns a and b.
From the factor theorem,
\[ \begin{align}
p(2) &= (2)^3 + a(2)^2 + b(2) - 4 \\
0 &= 8 + 4a + 2b - 4 \\
2a + b &= -2 \quad \quad (1)
\end{align} \ \]
From the factor theorem again,
\[ \begin{align}
p(-2) &= (-2)^3 + a(-2)^2 + b(-2) - 4 \\
0 &= -8 + 4a - 2b - 4 \\
2a - b &= 6 \quad \quad (2)
\end{align} \ \]
Eliminate b to find a first.
Adding (1) and (2):
\[ \begin{align}
4a &= 4 \\
a &= 1
\end{align} \ \]
from (1),
\[ \begin{align}
2 - b &= 6 \\
b &= -4
\end{align} \ \]
So
\[ p(x) = x^3 + x^2 - 4x - 4 \]
Express p(x) as a product of three linear factors.
\[ \begin{align}
x^3 + x^2 - 4x - 4 &= (x - 2)(x + 2)(x + k) \\
\text{So }\ -4 &= -4k \\
k &= 1
\end{align} \ \]
And
\[ x^3 + x^2 - 4x - 4 = (x - 2)(x + 2)(x + 1) \]
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