Algebra II Chapter 5 Review Notes

Chapter 5 Quest 1 Review Topics

  • Overview of Factoring

    • General Concepts: All Rules of Factoring

    • Greatest Common Factor (GCF)

      • The GCF is the largest factor that divides a set of numbers without leaving a remainder.

      • To find the GCF, list the factors of each number and identify the largest factor they have in common.

    • Trinomial Factoring

      • Involves expressing a trinomial in the form $ax^2 + bx + c$ as a product of two binomials: $(px + q)(rx + s)$.

      • The product needs to yield the original trinomial.

    • Guess & Check or Splitting the Middle Term

      • A method to factor a trinomial by guessing two numbers that multiply to $ac$ and add to $b$.

    • Difference of Two Squares

      • The formula is $a^2 - b^2 = (a + b)(a - b)$.

      • This states that the difference between two squares can be factored into the product of the sum and difference of the square roots of those terms.

    • Sum of Cubes

      • The formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

    • Difference of Cubes

      • The formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

  • Long Division of Polynomials

    • Used to divide a polynomial by another polynomial where the degree of the divisor is less than or equal to that of the dividend. The process involves repeated subtraction of multiples of the divisor from the dividend.

  • Synthetic Division

    • A simplified form of polynomial long division, this technique is used when dividing a polynomial by a linear binomial of the form $x - c$.

    • The coefficients of the polynomial are placed in a row; then, calculations are performed sequentially to find the quotient.

  • Finding All Roots

    • Procedure:

    1. Find factors of the leading coefficient and of the constant term of the polynomial.

    2. List all possible rational roots using the Rational Root Theorem.

    3. Test each rational root to see if it satisfies the polynomial equation.

      • If a root works, use synthetic or long division to divide the polynomial by $(x - ext{root})$.

    4. Factor the polynomial if possible, or repeat the process with the resulting polynomial.

    5. Set each factor equal to 0 to find the roots: If $f(x)=0$, then the roots are $x= ext{root}$.

  • Standard Form

    • The standard form of a polynomial is expressed as
      p(x)=axn+bxn−1+ext…+cp(x) = ax^n + bx^{n-1} + ext{…} + c

    • Every polynomial should be ordered from the highest degree term to the lowest degree term.

    • To find the polynomial from its roots, follow these steps:

    1. Set each root equal to x: If the roots are $r1, r2, ext{and} r3$, then the factors are $(x - r1)(x - r2)(x - r3)$.

    2. Multiply all factors together to yield the standard polynomial.

    3. Ensure the polynomial includes $y = f(x)$ if necessary.

Exercises in Factoring and Division

  • Factor Completely

    • Instruction: If the polynomial is not factorable, state “prime.”

  • Divide Using Long Division

    • Provide challenging polynomials for division practice.

  • Divide Using Synthetic Division

    • Suggested polynomial divisions to practice synthetic division with coefficients given.

  • Finding All Roots (Real or Imaginary)

    • Solve provided polynomial equations to find all roots. Use factoring, synthetic division, or numerical methods as applicable.

Additional Exercises

  • Writing Polynomials in Standard Form

    • Given sets of roots, write the associated polynomials in standard form with integral coefficients.

    • Specific roots to consider (e.g., roots 3, -4, etc.).

    • Example: If the roots are given as $3, -1$ then the polynomial can be formed as $(x - 3)(x + 1)$.