Saxon Math Course 2 - Lesson 115: Factoring Algebraic Expressions

This lesson covers the principles and methods for factoring algebraic expressions, including monomials and polynomials.
Lesson Number: 115.
Components included: Power Up (Facts, Mental Math, Problem Solving), New Concepts, Examples, Practice Set, and Written Practice.

Linear Addition and Subtraction: * $6x + 2x = 8x$ * $6x - 2x = 4x$ * $9xy + 3xy = 12xy$ * $9xy - 3xy = 6xy$ * $x + y + x = 2x + y$ * $x + y - x = y$ * $3x + x + 3 = 4x + 3$ * $3x - x - 3 = 2x - 3$
Multiplication and Exponentiation: * $(6x)(2x) = 12x^2$ * $(9xy)(3xy) = 27x^2y^2$ * $(x)(y)(-x) = -x^2y$ * $(3x)(-x)(-3) = 9x^2$
Division and Fractions: * $\frac{6x}{2x} = 3$ * $\frac{9xy}{3xy} = 3$ * $\frac{xy}{xy} = 1$ * $\frac{x}{xy} = \frac{1}{y}$ * $\frac{(2x)(8xy)}{4y} = 4x^2$

a. Positive/Negative: $10 - 2$
b. Scientific Notation: $(4 \times 10^8) \div (4 \times 10^8)$
c. Ratio: $\frac{1.44}{1.2} = \frac{1.2}{g}$
d. Measurement: Convert $250\,\text{cm}$ to $\text{m}$ .
e. Fractional Parts: $\frac{2}{3}$ of $\$1200$
f. Fractional Parts: $\$1200$ reduced $\frac{1}{13}$
g. Primes/Composites: Determine which number does not belong in the list: $12, 15, 19, 21$ .
h. Money: Determine how many cents a nickel is less than the sum of $3$ dimes and $3$ quarters.

Task: Determine what digit every letter represents in the following multiplication problems. Each letter represents the same missing digit in every problem.
Problem 1: * $ABCDEF \times B = 248,712$
Problem 2: * $BCGHAB \times D = 746,136$
Problem 3: * $HCFADF \times C = 2,984,544$

Factoring is the process of breaking down a term into its prime factors and variable components.
Example 1: Factor the monomial $12a^2b^3c$ . * Solution Process: 1. Factor the numerical coefficient: $12$ is factored into $(2)(2)(3)$ . 2. Factor the variable component: $a^2b^3c$ is factored into $aabbbc$ . * Final Result: $12a^2b^3c = 2 \cdot 2 \cdot 3 \cdot a \cdot a \cdot b \cdot b \cdot b \cdot c$

Factoring a polynomial involves identifying the greatest common factor (GCF) of all terms in the expression.
Example 2: Factor the trinomial $6a^2b + 4ab^2 + 2ab$ . * Step 1: Identify the GCF visually or by factorization. Each term in this expression contains factors of $2$ , $a$ , and $b$ . Therefore, the GCF is $2ab$ . * Step 2: Divide each individual term of the original trinomial by the GCF ( $2ab$ ) to determine the remaining terms. * $\frac{6a^2b}{2ab} = 3a$ * $\frac{4ab^2}{2ab} = 2b$ * $\frac{2ab}{2ab} = 1$ * Step 3: Combine the GCF and the remaining trinomial. * Result: The GCF is $2ab$ and the remaining trinomial is $3a + 2b + 1$ . The factored expression is $2ab(3a + 2b + 1)$ .

Factor the following algebraic expressions: * a. $8m^2n$ * b. $12mn^2$ * c. $18x^32$ * d. $8m^2n + 12mn^2$ * e. $8xy^2 - 4xy$ * f. $6a^2b^3 + 9a^3b^2 + 3a^2b^2$
Classify (Problem g): What type of polynomial is the expression in problem f?

1. $\$210$ ; Sample: The deposit earns $\$280$ in one year, so in $9$ months, the deposit earns of $\$280$ , or $\$210$ .
2. a. . b. $25\%$ c. $1$ to $3$
3. $348$
4. bottle costs more; $1$ cents per ounce more
5. $25$ minutes
6. $70$ robins
7. $40$
8. $\$8316$
9. $\$48$