6.10 Integrating Functions Using Long Division and Completing the Square

• Steps of long division for polynomials

  1️⃣ Divide the leading term of the numerator by the leading term of the denominator

  2️⃣ Multiply the result by the denominator and subtract from the numerator

  3️⃣ Bring down the next term of the numerator

  4️⃣ Repeat steps 1-3 until the degree of the numerator is less than the degree of the denominator
• To integrate a function using long division, the last step involves integrating the resulting quotient
• When dividing $3x^{2} - 2x +$$5$ by $x +$$2$, the quotient is 3x - 8
• In integration by long division, the goal is to simplify the integrand
• The simplified expression after dividing \frac{x^2}{x+1}</latex> using long division is $x - 1 +$$\frac{1}{x + 1}$


  True
• What technique is used to simplify rational functions for integration?

  Long division
• What is the integral of $\frac{1}{x + 1}$?


  $\ln|x +$$1|$
• Steps for long division of polynomials

  1️⃣ Divide the leading term of the numerator by the leading term of the denominator

  2️⃣ Multiply the result by the denominator and subtract from the numerator

  3️⃣ Bring down the next term of the numerator

  4️⃣ Repeat steps 1-3 until the degree of the numerator is less than the degree of the denominator
• Integrating rational functions using long division involves dividing the numerator by the denominator first.


  True
• Long division of polynomials requires repeating steps until the degree of the numerator is less than the degree of the denominator
• After performing long division on $\frac{x^{2}}{x + 1}$, the remainder is 1
• What is the first step in polynomial long division?

  Divide leading terms
• The process of polynomial long division continues until the degree of the numerator is less than the degree of the denominator.


  True
• Match the steps in long division with their counterparts in integration by long division:

  Divide leading terms ↔️ Divide numerator by denominator

  Multiply and subtract ↔️ Multiply result by denominator, subtract

  Bring down next term ↔️ Integrate the result from step 2

  Repeat steps 1-3 ↔️ Repeat steps 1-3
• In polynomial long division, you multiply the result by the denominator
• What is the first step in integrating rational functions using long division?

  Divide numerator by denominator
• The final step in integrating rational functions using long division is to integrate the result from step 2.

  True
• What is the first step in long division for polynomials?

  Divide leading terms
• The process of polynomial long division continues until the degree of the numerator is less than the degree of the denominator.


  True
• In integrating rational functions using long division, you subtract the result from the numerator
• Match the steps in long division with their counterparts in integration by long division:

  Divide leading terms ↔️ Divide numerator by denominator

  Multiply and subtract ↔️ Multiply result by denominator, subtract

  Bring down next term ↔️ Integrate the result from step 2

  Repeat steps 1-3 ↔️ Repeat steps 1-3
• In polynomial long division, you multiply the result by the denominator
• The process of polynomial long division continues until the degree of the numerator is less than the degree of the denominator.


  True
• In integrating rational functions using long division, you subtract the result from the numerator
• Match the steps in long division with their counterparts in integration by long division:

  Divide leading terms ↔️ Divide numerator by denominator

  Multiply and subtract ↔️ Multiply result by denominator, subtract

  Bring down next term ↔️ Integrate the result from step 2

  Repeat steps 1-3 ↔️ Repeat steps 1-3
• In polynomial long division, you multiply the result by the denominator
• What is the first step in integrating rational functions using long division?

  Divide numerator by denominator
• The final step in integrating rational functions using long division is to integrate the result from step 2.


  True
• In completing the square, you divide the coefficient of x</latex> by 2
• Match the steps in completing the square with their descriptions:

  Factor out $x^{2}$ coefficient ↔️ Ensure the $x^{2}$ term has a coefficient of 1

  Calculate and add/subtract $(b / 2)^{2}$ ↔️ Create a perfect square trinomial

  Group and simplify ↔️ Rewrite the expression as a perfect square plus a constant
• When completing the square for $x^{2} +$$6x +$$1$, the expression becomes (x + 3)^{2} - 8
• If the coefficient of $x^{2}$ is not 1, it must be factored out before completing the square.


  True
• What term is added and subtracted to complete the square for $ax^{2} +$$bx +$$c$?


  $(\frac{b}{2a})^{2}$
• What is the first step to complete the square for a quadratic expression of the form $ax^{2} +$$bx +$$c$?


  Factor out the coefficient of $x^{2}$
• For the expression $x^{2} +$$8x +$$10$, what value is added and subtracted to complete the square?


  16
• Steps for performing long division for polynomials

  1️⃣ Divide the leading term of the numerator by the leading term of the denominator

  2️⃣ Multiply the result by the denominator and subtract from the numerator

  3️⃣ Bring down the next term of the numerator

  4️⃣ Repeat steps 1-3 until the degree of the numerator is less than the degree of the denominator
• In integration by long division, the first step is to divide the numerator by the denominator.


  True
• Match the steps of long division with their corresponding actions:

  Divide leading terms ↔️ Divide numerator by denominator

  Multiply and subtract ↔️ Multiply result by denominator, subtract

  Bring down next term ↔️ Integrate the result from step 2
• Long division for polynomials involves dividing the leading term of the numerator by the leading term of the denominator.


  True
• Integrating a rational function using long division involves integrating the remainder after division.


  True