Sure, let's find the quotient of the given expression step-by-step.
Given the expression:
[tex]\[ \frac{x^4 - 5x^2 + 2x + 1}{x^2 + x + 3} \][/tex]
### Step 1: Arrange the Polynomial Division
We write the expression in the form of polynomial long division:
[tex]\[ x^4 - 5x^2 + 2x + 1 \div x^2 + x + 3 \][/tex]
### Step 2: Divide the Leading Terms
We start by dividing the leading term of the numerator [tex]\( x^4 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{x^4}{x^2} = x^2 \][/tex]
This gives us the first term of the quotient [tex]\( x^2 \)[/tex].
### Step 3: Multiply and Subtract
Next, multiply [tex]\( x^2 \)[/tex] by the entire denominator [tex]\( x^2 + x + 3 \)[/tex]:
[tex]\[ x^2 \cdot (x^2 + x + 3) = x^4 + x^3 + 3x^2 \][/tex]
Subtract this result from the original numerator:
[tex]\[ (x^4 - 5x^2 + 2x + 1) - (x^4 + x^3 + 3x^2) \][/tex]
[tex]\[ = -x^3 - 8x^2 + 2x + 1 \][/tex]
### Step 4: Repeat the Process
Now, divide the new leading term [tex]\( -x^3 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{-x^3}{x^2} = -x \][/tex]
This gives us the next term of the quotient [tex]\( -x \)[/tex].
### Step 5: Multiply and Subtract Again
Multiply [tex]\( -x \)[/tex] by the entire denominator:
[tex]\[ -x \cdot (x^2 + x + 3) = -x^3 - x^2 - 3x \][/tex]
Subtract this from the new numerator:
[tex]\[ (-x^3 - 8x^2 + 2x + 1) - (-x^3 - x^2 - 3x) \][/tex]
[tex]\[ = -7x^2 + 5x + 1 \][/tex]
### Step 6: Continue the Process
Next, divide the new leading term [tex]\( -7x^2 \)[/tex] by the leading term [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{-7x^2}{x^2} = -7 \][/tex]
This gives us the final term of the quotient [tex]\( -7 \)[/tex].
### Step 7: Multiply and Subtract One Last Time
Multiply [tex]\( -7 \)[/tex] by the entire denominator:
[tex]\[ -7 \cdot (x^2 + x + 3) = -7x^2 - 7x - 21 \][/tex]
Subtract this from the current numerator:
[tex]\[ (-7x^2 + 5x + 1) - (-7x^2 - 7x - 21) \][/tex]
[tex]\[ = 12x + 22 \][/tex]
At this point, the degree of the remainder [tex]\( 12x + 22 \)[/tex] is less than the degree of the denominator [tex]\( x^2 + x + 3 \)[/tex], so we stop here.
### Final Answer:
The quotient is:
[tex]\[ x^2 - x - 7 \][/tex]
And the remainder is:
[tex]\[ 12x + 22 \][/tex]
