Finding Constants Using Factor Theorem

Video Explanation

Question

If \(x+2\) is a factor of the polynomial

\[ f(x) = x^2 + ax + 2b \]

and

\[ a + b = 4, \]

then find the values of \(a\) and \(b\).

Options:

(a) \(a = 1,\; b = 3\)
(b) \(a = 3,\; b = 1\)
(c) \(a = -1,\; b = 5\)
(d) \(a = 5,\; b = -1\)

Solution

Step 1: Use the Factor Theorem

Since \(x + 2\) is a factor of \(f(x)\),

\[ f(-2) = 0 \]

Step 2: Substitute \(x = -2\)

\[ (-2)^2 + a(-2) + 2b = 0 \]

\[ 4 – 2a + 2b = 0 \]

Divide throughout by 2:

\[ 2 – a + b = 0 \Rightarrow b = a – 2 \]

Step 3: Use the Given Condition

Given:

\[ a + b = 4 \]

Substitute \(b = a – 2\):

\[ a + (a – 2) = 4 \]

\[ 2a = 6 \Rightarrow a = 3 \]

Then,

\[ b = 4 – a = 1 \]

Conclusion

The correct values are:

\[ \boxed{a = 3,\; b = 1} \]

Hence, the correct option is (b).