Of course! Let's go through a detailed, step-by-step solution for dividing the polynomial [tex]\(x^2 - 3x - 28\)[/tex] by [tex]\(x - 7\)[/tex].
### Step 1: Set up the Division
We are dividing [tex]\(x^2 - 3x - 28\)[/tex] by [tex]\(x - 7\)[/tex].
[tex]\[
\frac{x^2 - 3x - 28}{x - 7}
\][/tex]
### Step 2: Divide the Leading Terms
First, divide the leading term of the numerator ([tex]\(x^2\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]).
[tex]\[
\frac{x^2}{x} = x
\][/tex]
So, [tex]\(x\)[/tex] is the first term of the quotient.
### Step 3: Multiply and Subtract
Next, multiply the divisor [tex]\(x - 7\)[/tex] by the quotient term [tex]\(x\)[/tex]:
[tex]\[
(x - 7) \cdot x = x^2 - 7x
\][/tex]
Now, subtract this result from the original polynomial:
[tex]\[
(x^2 - 3x - 28) - (x^2 - 7x) = x^2 - 3x - 28 - x^2 + 7x = 4x - 28
\][/tex]
### Step 4: Repeat the Process
Now, we repeat the process with the new polynomial [tex]\(4x - 28\)[/tex]. Divide the leading term [tex]\(4x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{4x}{x} = 4
\][/tex]
So, [tex]\(4\)[/tex] is the next term of the quotient.
### Step 5: Multiply and Subtract Again
Multiply the divisor [tex]\(x - 7\)[/tex] by [tex]\(4\)[/tex]:
[tex]\[
(x - 7) \cdot 4 = 4x - 28
\][/tex]
Subtract this from the polynomial [tex]\(4x - 28\)[/tex]:
[tex]\[
(4x - 28) - (4x - 28) = 0
\][/tex]
### Step 6: Conclusion
No remainder is left. The quotient of the division is [tex]\(x + 4\)[/tex] and the remainder is [tex]\(0\)[/tex].
Hence, the division of [tex]\(\left(x^2 - 3x - 28\right)\)[/tex] by [tex]\(\left(x - 7\right)\)[/tex] results in:
[tex]\[
(x + 4, \text{ remainder } 0)
\][/tex]
So, the quotient is [tex]\(x + 4\)[/tex] and the remainder is [tex]\(0\)[/tex].
