Question

If $x + 1$ is a factor of $p(x) = 4x^2 + 3x + k$, then the value of $k$ is

• A. 8
• B. -8
• C. 1
• D. -1

Hint:

Whenever a linear expression like $x-a$ or $x+b$ is given as a factor, directly apply the Factor Theorem: substitute the root value into the polynomial and equate it to zero.

Answer 1

The Correct Option is C

Step 1: Use the factor theorem.}
If $x + 1$ is a factor of the polynomial $p(x)$, then by the Factor Theorem, we must have: \[ p(-1) = 0 \] This means that when we substitute $x = -1$ into the polynomial, the result must be zero.

Step 2: Substitute $x = -1$ into the polynomial.}
Given: \[ p(x) = 4x^2 + 3x + k \] Now put $x = -1$: \[ p(-1) = 4(-1)^2 + 3(-1) + k \] \[ p(-1) = 4(1) - 3 + k \] \[ p(-1) = 1 + k \] Since $x + 1$ is a factor, we know: \[ 1 + k = 0 \]

Step 3: Solve for $k$.}
From \[ 1 + k = 0 \] we get: \[ k = -1 \] So, the required value of $k$ is: \[ -1 \]

Step 4: Compare with the given options.}

• (A) 8: Incorrect. This does not make $p(-1) = 0$.
• (B) -8: Incorrect. This also does not satisfy the factor condition.
• (C) 1: Incorrect. This would give $p(-1) = 2$, not 0.
• (D) -1: Correct. This makes $p(-1) = 0$, so $x+1$ becomes a factor.

Hence, the correct answer is $-1$.
Final Answer:} $-1$.