MAT 101 Final Exam Study Guide: Factoring, Rational Expressions, Radicals, and Quadratic Equations

Factoring Polynomials

Greatest Common Factor (GCF)

The greatest common factor (GCF) of a polynomial is the largest expression that divides each term of the polynomial without remainder.

• Finding the GCF: Identify the largest numerical and variable factors common to all terms.

• Example: For $6x^3 + 9x^2$, the GCF is $3x^2$.

• Factoring out the GCF: Write the polynomial as the product of the GCF and the remaining terms. Example: $6x^3 + 9x^2 = 3x^2(2x + 3)$

Factoring by Grouping

Factoring by grouping is used for polynomials with four or more terms. Group terms to factor out common factors, then factor the resulting expression.

• Example: $x^3 + 3x^2 + 2x + 6$ Group: $(x^3 + 3x^2) + (2x + 6)$ Factor: $x^2(x + 3) + 2(x + 3)$ Final: $(x^2 + 2)(x + 3)$

Factoring Trinomials

Trinomials of the form $x^2 + bx + c$ can often be factored into two binomials.

• Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$

• Method: Find two numbers that multiply to $c$ and add to $b$.

Perfect Squares and Difference of Squares

• Perfect Square Trinomials: Expressions like $x^2 + 6x + 9$ factor as $(x + 3)^2$.

• Difference of Squares: Expressions like $x^2 - 9$ factor as $(x + 3)(x - 3)$.

Solving Quadratic Equations

Zero-Factor Property

If a product of factors equals zero, then at least one factor must be zero.

• Example: If $(x + 2)(x - 3) = 0$, then $x + 2 = 0$ or $x - 3 = 0$.

• Solutions: $x = -2$ or $x = 3$

Solving Quadratics by Completing the Square

Completing the square transforms a quadratic equation into a perfect square trinomial.

• Example: $x^2 + 6x + 5 = 0$ Move constant: $x^2 + 6x = -5$ Add $(\frac{6}{2})^2 = 9$: $x^2 + 6x + 9 = 4$ Factor: $(x + 3)^2 = 4$ Solve: $x + 3 = \pm 2$ $x = -1$ or $x = -5$

Quadratic Formula

The quadratic formula solves any quadratic equation $ax^2 + bx + c = 0$:

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: $x^2 + 5x + 6 = 0$ $a = 1$, $b = 5$, $c = 6$ $x = \frac{-5 \pm \sqrt{25 - 24}}{2}$ $x = \frac{-5 \pm 1}{2}$ $x = -2$ or $x = -3$

Polynomial Equations in Applications

Word problems may require setting up and solving polynomial equations based on real-world scenarios.

• Example: If the area of a rectangle is $x^2 + 5x + 6$, and the length is $x + 2$, find the width. Set up: $\text{Area} = \text{Length} \times \text{Width}$ $x^2 + 5x + 6 = (x + 2) \times \text{Width}$ $\text{Width} = \frac{x^2 + 5x + 6}{x + 2} = x + 3$

Rational Expressions

Reducing Rational Expressions

A rational expression is a fraction with polynomials in the numerator and denominator. Reduce by factoring and canceling common factors.

• Example: $\frac{x^2 + 5x + 6}{x + 2} = \frac{(x + 2)(x + 3)}{x + 2} = x + 3$

Multiplying and Dividing Rational Expressions

• Multiply: Multiply numerators and denominators, then reduce. Example: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

• Divide: Multiply by the reciprocal. Example: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Adding and Subtracting Rational Expressions

• Find a common denominator before adding or subtracting.

• Example: $\frac{1}{x} + \frac{2}{x} = \frac{1 + 2}{x} = \frac{3}{x}$

Simplifying Complex Fractions

A complex fraction has fractions in the numerator, denominator, or both. Simplify by finding a common denominator or multiplying by the reciprocal.

• Example: $\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x}}$ Find common denominators and simplify.

Radical Expressions

Meaning of Radical Expressions

A radical expression involves roots, such as square roots or cube roots.

• Example: $\sqrt{16} = 4$, $\sqrt[3]{27} = 3$

Radicals of Perfect Squares and Cubes

• Perfect Square: $\sqrt{9} = 3$

• Perfect Cube: $\sqrt[3]{27} = 3$

Radicals as Exponential Expressions

Radicals can be rewritten using exponents:

• $\sqrt[n]{x} = x^{1/n}$

• Example: $\sqrt{16} = 16^{1/2} = 4$

Operations with Radical Expressions

• Add/Subtract: Combine like radicals. Example: $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$

• Multiply: Multiply coefficients and radicands. Example: $\sqrt{2} \times \sqrt{3} = \sqrt{6}$

Solving Equations with Radicals

• Isolate the radical and then raise both sides to the appropriate power to eliminate the radical.

• Example: $\sqrt{x} = 5$ Square both sides: $x = 25$

Quadratic Equations with Radicals and Completing the Square

• Quadratic equations may involve radicals or require completing the square for solution.

• Example: $x^2 = 9$ Take square root: $x = \pm 3$

Word Problems Involving Rates

Problems involving rates often require setting up equations based on the relationship: $\text{Rate} = \frac{\text{Distance}}{\text{Time}}$.

• Example: If a car travels 60 miles in 2 hours, its rate is $\frac{60}{2} = 30$ miles per hour.

Exam Preparation Tips

• Redo worksheets and practice problems without notes.

• Review old homework, labs, and quizzes.

• Seek help from instructors or tutors for difficult topics.