Algebra: Factoring Review

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Greatest Common Factor

• To find the GCF of $15x^5y^3$, $3x^8y$, and $27x^4y^4$, identify the smallest power of each common variable and the GCF of the coefficients
• The GCF is $3x^3y$

Factoring Expressions

• $(x-3)(x+5) + (x-3)(x+8)$ can be factored by taking out the common factor $(x-3)$, resulting in $(x-3)(x+5+x+8) = (x-3)(2x+13)$

Factoring Expressions

• $3-6x-4x+8x^2$ can be factored by grouping
• $3(1-2x) - 4x(1-2x)$
• $(1-2x)(3-4x)$

Factoring Quadratics

• $x^2+5x+6$ factors to $(x+2)(x+3)$
• Finding two numbers that add up to 5 and multiply to 6 helps identify +2 and +3

Factoring Quadratics

• $x^2-49$ is a difference of squares and factors to $(x+7)(x-7)$

Factoring Quadratics

• $3x^2 - 243$ can be simplified by factoring out 3: $3(x^2-81)$, further factored to $3(x+9)(x-9)$

Factoring Quadratics

• To factor $x^2-3x+10$, find two numbers that add to -3 and multiply to 10, resulting in $(x+2)(x-5)$

Functions - domain and interval notation

• Given $f(x) = \frac{3x-7}{4x-9}$, the domain is all real numbers except where the denominator is zero
• Set $4x-9=0$ to find the excluded value: $x = \frac{9}{4}$
• Therefore, the domain is all real numbers $x \neq \frac{9}{4}$
• In interval notation: $(-\infty, \frac{9}{4}) \cup (\frac{9}{4}, \infty)$

Simplifying Fractions

• Simplifying $\frac{15}{49} \cdot \frac{7}{10}$ involves canceling common factors, resulting in $\frac{3}{7} \cdot \frac{1}{2} = \frac{3}{14}$

Simplifying Rational Expressions

• $\frac{(x+5)(x-1)}{(x-1)(x+2)}$ simplifies to $\frac{x+5}{x+2}$ by canceling the common factor $(x-1)$

Dividing Rational Expressions

• $\frac{(3x-5)(x-3)}{(x+7)(x-3)}$ divided by $\frac{(x-5)(x+3)}{(x+7)(x-5)}$ simplifies by multiplying by the reciprocal, canceling common factors
• Resulting in $\frac{3x-5}{x+3}$

Simplifying Rational Expressions

• $\frac{15x+6}{14x} \cdot \frac{25x+10}{7}$ simplifies to $\frac{3(5x+2)}{14x}\cdot \frac{5(5x+2)}{7} = \frac{15(5x+2)}{7}$

Least Common Denominator

• To find the least common denominator (LCD) for $\frac{1}{3x}$ and $\frac{1}{x^2}$, identify the highest power of each unique factor in the denominators
• The LCD is $3x^2$

Adding Rational Expressions

• $\frac{5x}{7} + \frac{11y}{7}$ can be combined directly since they have a common denominator, resulting in $\frac{5x+11y}{7}$

Subtracting Rational Expressions

• $\frac{x}{3x-2y} - \frac{y}{2y-3x}$ can be rewritten as $\frac{x}{3x-2y} + \frac{y}{3x-2y}$ by factoring out a -1 from the second denominator
• The result is $\frac{x+y}{3x-2y}$

Solving Rational Equations

• To solve $\frac{1}{x+5} - \frac{1}{2x-3} = 0$, first identify the values of x that would make the denominators zero
• $x = -5, x=\frac{3}{2}$
• These values are excluded from the domain, so the domain is all real numbers $x \neq -5, \frac{3}{2}$

Solving Rational Equations

• Solve $\frac{2}{5x+3} + \frac{x}{2} = \frac{6}{15x+9}$
• Multiplying both sides by the LCD, $2(5x+3)$ yields $2(2) + x(5x+3) = 2(2)$
• $5x^2 + 3x = 0$
• Factoring out x gives $x(5x+3) = 0$
• Solutions are $x=0$ and $x=-\frac{3}{5}$

Simplifying Radicals

• $\sqrt[5]{-64} = -2\sqrt[5]{2}$

Simplifying Radicals

• $\sqrt[7]{(-11)^7} = -11$ because the seventh root cancels out the seventh power

Simplifying Radicals

• $3^{\frac{5}{3}}$ can be written as $\sqrt[3]{3^5} = \sqrt[3]{3^3 \cdot 3^2} = 3\sqrt[3]{3^2}$

Rational Exponents

• $(\frac{x^{\frac{1}{3}}y^{\frac{2}{3}}}{x^2})^{15}$
• $x^{5-2}y^9 = x^3y^9$

Radical Multiplication

• Cube root of $5 \cdot$ cube root of $19$ = cube root of $95$

Radical Simplification

• $-3 \sqrt[3]{\frac{64}{x^6}}$ simplifies to $-\frac{4}{x^2}$

Simplifying Radical Expressions

• $3\sqrt[3]{27x^9y^{21}} = 3x^3y^7$

Combining Radicals

• $-5\sqrt{3} + 3\sqrt{27} = -5\sqrt{3} + 9\sqrt{3} = 4\sqrt{3}$

Distributing Radicals

• $\sqrt{5}(\sqrt{15}-\sqrt{3}) = \sqrt{75} - \sqrt{15}$ simplifies to $5\sqrt{3} - \sqrt{15}$