6.10 Integrating Functions Using Long Division and Completing the Square

• Rational functions are functions of the form $f(x) =$$\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials
• In long division of rational functions, the numerator and divisor must be arranged in ascending order of exponents.

  False
• Steps to divide rational functions using long division

  1️⃣ Arrange numerator and divisor in descending order of exponents

  2️⃣ Divide the leading term of the numerator by the leading term of the divisor

  3️⃣ Multiply the quotient by the entire divisor

  4️⃣ Subtract this product from the numerator

  5️⃣ Bring down the next term

  6️⃣ Repeat until all terms are used

  7️⃣ The final result is the remainder
• In long division of rational functions, the final result after all steps is called the remainder
• Match the division problem with its result:

  \frac{x^{2} + 3x + 5}{x + 1}</latex> ↔️ $x +$$2 +$$\frac{3}{x + 1}$

  \frac{x^{3} + 2x^{2} - 3x - 4}{x + 2} ↔️ $x^{2} - 3 +$$\frac{2}{x + 2}$

  \frac{2x^{3} +$3x^{2} +$ x - 2}{x + 1} ↔️ $2x^{2} +$$x - 1 +$$\frac{ - 2}{x + 1}$
• What is the simplified result of \frac{x^{2} +$3x +$ 5}{x + 1} after long division?


  $x +$$2 +$$\frac{3}{x + 1}$
• To perform long division on rational functions, the polynomials must be arranged in descending order of exponents
• The quotient and remainder are the final results of long division for rational functions.
• Why is simplifying rational functions after long division useful?

  Easier to integrate
• After long division, a rational function can be simplified as \frac{P(x)}{Q(x)} = A(x) + \frac{R(x)}{Q(x)}</latex>, where $A(x)$ is the quotient
• What is the simplified form of \frac{x^{3} + 2x^{2} - 3x - 4}{x + 2} after long division?


  $x^{2} - 3 +$$\frac{2}{x + 2}$
• Match the steps of long division with their descriptions:

  Arrange polynomials ↔️ Order by descending exponents

  Divide leading terms ↔️ Find the first term of the quotient

  Multiply quotient ↔️ Multiply the divisor by the quotient

  Subtract product ↔️ Subtract the result from the numerator
• What is the first step in dividing two polynomials using long division?

  Divide the leading terms
• The final results of long division are the quotient and the remainder
• The example in line 8 demonstrates the division of $x^{3} +$$2x^{2} - 3x - 4$ by $x +$$2$.
• What is the simplified result of dividing \frac{x^{3} + 2x^{2} - 3x - 4}{x + 2}</latex>?

  $x^{2} - 3 +$$\frac{2}{x + 2}$
• What are the components of a rational function?

  Two polynomials
• In simplifying rational functions, the division result is expressed as the quotient plus the remainder over the divisor
• The remainder and divisor of a rational function must have common factors to simplify it further.
• Steps to complete the square for ax^{2} + bx + c</latex>

  1️⃣ Factor out the leading coefficient a

  2️⃣ Add and subtract $(\frac{b}{2a})^{2}$

  3️⃣ Rewrite as a squared binomial

  4️⃣ Simplify the expression
• Match the quadratic form with its description:

  Standard Form ↔️ $ax^{2} +$$bx +$$c$

  Completed Square Form ↔️ $a(x + h)^{2} +$$k$
• To complete the square for $2x^{2} +$$8x +$$5$, you first factor out 2
• What is the completed square form of $2x^{2} +$$8x +$$5$?


  $2(x + 2)^{2} - 3$
• What is the first step in applying completing the square to a quadratic expression?

  Factor out the leading coefficient
• In the completed square form $a(x + h)^{2} +$$k$, $h =$$\frac{b}{2a}$.
• When completing the square for $3x^{2} +$$12x +$$11$, the first step is to factor out 3
• What is the completed square form of $3x^{2} +$$12x +$$11$?


  $3(x + 2)^{2} - 1$
• Match the concept with its description:

  Rational Function ↔️ $\frac{P(x)}{Q(x)}$

  Long Division ↔️ Dividing polynomials

  Quotient ↔️ Result of division

  Remainder ↔️ Leftover after division
• To divide rational functions using long division, arrange both polynomials in descending order of exponents
• What is the second step in dividing rational functions using long division?

  Divide leading terms
• What is the form of a rational function?

  $f(x) =$$\frac{P(x)}{Q(x)}$
• In a rational function, $P(x)$ and $Q(x)$ are both polynomials
• The first step in dividing rational functions using long division is to arrange the polynomials in ascending order of exponents.

  False
• Steps to divide rational functions using long division

  1️⃣ Arrange polynomials in descending order of exponents

  2️⃣ Divide the leading term of $P(x)$ by the leading term of $Q(x)$

  3️⃣ Multiply $Q(x)$ by the first term of the quotient

  4️⃣ Subtract this product from $P(x)$

  5️⃣ Bring down the next term from $P(x)$

  6️⃣ Repeat steps 2-5 until no terms remain

  7️⃣ Identify the quotient and remainder
• What is the simplified form of a rational function after long division?

  $\frac{P(x)}{Q(x)} =$$A(x) +$$\frac{R(x)}{Q(x)}$
• Simplifying a rational function after long division makes it easier to integrate.
• After expressing a rational function in simplified form, check for common factors between $R(x)$ and $Q(x)$Q(x)</latex>
• Steps to simplify a rational function after long division

  1️⃣ Express as quotient and remainder

  2️⃣ Check for common factors between $R(x)$ and $Q(x)$

  3️⃣ Apply additional algebraic techniques if needed
• What is the purpose of completing the square for a quadratic expression?

  To transform it into $a(x + h)^{2} +$$k$
• In the completed square form $a(x + h)^{2} +$$k$, $h$ is equal to \frac{b}{2a}