`4x^(2)+kx+5` is divisible by `x-1`. The expression is also divisible by

Options

A. `4x-5`
B. `x-5`
C. `4x-1`
D. `4x+5`

Answer

To solve the problem step by step, we need to determine the value of \( k \) such that the polynomial \( 4x^2 + kx + 5 \) is divisible by \( x - 1 \).

### Step 1: Use the Factor Theorem
Since \( 4x^2 + kx + 5 \) is divisible by \( x - 1 \), by the Factor Theorem, substituting \( x = 1 \) into the polynomial should yield 0.

### Step 2: Substitute \( x = 1 \)
Substituting \( x = 1 \) into the polynomial:
\[ 4(1)^2 + k(1) + 5 = 0 \]
This simplifies to:
\[ 4 + k + 5 = 0 \]

### Step 3: Solve for \( k \)
Combine the constants:
\[ k + 9 = 0 \]
Now, isolate \( k \):
\[ k = -9 \]

### Step 4: Write the polynomial with the found \( k \)
Now that we have \( k \), we can rewrite the polynomial:
\[ 4x^2 - 9x + 5 \]

### Step 5: Factor the polynomial
Next, we will factor \( 4x^2 - 9x + 5 \). We look for two numbers that multiply to \( 4 \times 5 = 20 \) and add to \( -9 \). The numbers are \( -4 \) and \( -5 \).

Rewriting the polynomial:
\[ 4x^2 - 4x - 5x + 5 = 0 \]

### Step 6: Group the terms
Grouping the first two and the last two terms:
\[ (4x^2 - 4x) + (-5x + 5) = 0 \]

### Step 7: Factor by grouping
Factoring out the common terms:
\[ 4x(x - 1) - 5(x - 1) = 0 \]
Now, factor out \( (x - 1) \):
\[ (x - 1)(4x - 5) = 0 \]

### Conclusion
Thus, the polynomial \( 4x^2 - 9x + 5 \) is factored as \( (x - 1)(4x - 5) \). Therefore, the expression is also divisible by \( 4x - 5 \).

### Final Answer
The expression is divisible by \( 4x - 5 \).

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