Sure, let's break down the division of the polynomial [tex]\( x^2 + 17x - 9 \)[/tex] by [tex]\( x + 3 \)[/tex] step by step.
### Step 1: Set Up the Division
We are dividing [tex]\( x^2 + 17x - 9 \)[/tex] by [tex]\( x + 3 \)[/tex]. This process is an example of polynomial long division.
### Step 2: Divide the Leading Terms
First, we look at the leading term of the dividend [tex]\( x^2 \)[/tex] and the leading term of the divisor [tex]\( x \)[/tex]. We ask, "What do we multiply [tex]\( x \)[/tex] by to get [tex]\( x^2 \)[/tex]?" The answer is [tex]\( x \)[/tex].
So, [tex]\( x \)[/tex] is the first term of our quotient. Write down [tex]\( x \)[/tex].
### Step 3: Multiply and Subtract
Now, we multiply [tex]\( x \)[/tex] by the divisor [tex]\( x + 3 \)[/tex]:
[tex]\[ x \cdot (x + 3) = x^2 + 3x \][/tex]
Next, we subtract this product from the original polynomial:
[tex]\[
(x^2 + 17x - 9) - (x^2 + 3x) = 14x - 9
\][/tex]
### Step 4: Repeat the Process
We repeat the process with the new polynomial [tex]\( 14x - 9 \)[/tex].
Again, divide the leading term of the new polynomial [tex]\( 14x \)[/tex] by the leading term of the divisor [tex]\( x \)[/tex]:
[tex]\[ 14x \div x = 14 \][/tex]
So, [tex]\( 14 \)[/tex] is the next term of our quotient. Write down [tex]\( 14 \)[/tex].
Multiply [tex]\( 14 \)[/tex] by the divisor:
[tex]\[ 14 \cdot (x + 3) = 14x + 42 \][/tex]
Subtract this product from the new polynomial:
[tex]\[
(14x - 9) - (14x + 42) = -51
\][/tex]
### Step 5: Combine the Quotient
Combining the terms we wrote down, the quotient is [tex]\( x + 14 \)[/tex].
### Step 6: Remainder
The remainder is the result from the last subtraction, which is [tex]\( -51 \)[/tex].
### Conclusion
So, when we divide [tex]\( x^2 + 17x - 9 \)[/tex] by [tex]\( x + 3 \)[/tex], the quotient is:
[tex]\[ x + 14 \][/tex]
and the remainder is:
[tex]\[ -51 \][/tex]
Thus, we can write:
[tex]\[
\frac{x^2 + 17x - 9}{x + 3} = x + 14 + \frac{-51}{x + 3}
\][/tex]
Or, more concisely:
[tex]\[
(x^2 + 17x - 9) \div (x + 3) = x + 14 \ \text{with a remainder of} \ -51
\][/tex]
