CLEP Algebra Flashcards

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What do you need to do to add or subtract fractions?

Get a common denominator

How do you multiply fractions?

Multiply numerators and multiply denominators

How do you divide by a fraction?

Multiply the entire term by the reciprocal fraction

Compute: (1/3) + (7/5) = ___

(26/15) Signup and view all the answers

Compute: [(3x-1) / (x^2-2)] + [1/(x-1)] = ___

(3x^2 - 4x + 2) / (x^3 - x^2 - 2x + 2) Signup and view all the answers

What is another way to write √x?

x^(1/2) Signup and view all the answers

(√x)^2 = ?

x Signup and view all the answers

Roots and powers undo each other.

True Signup and view all the answers

How do you multiply 2 terms with the same base but different exponents?

Add the exponents Signup and view all the answers

How do you multiply 2 terms with different bases but the same exponents?

Multiply the bases and raise this to the exponent Signup and view all the answers

How do you solve a term with an exponent, and that whole term is raised to another exponent?

Multiply the exponents Signup and view all the answers

How do you divide terms with exponents?

Subtract the exponents Signup and view all the answers

Simplify: [(4√9)^8] / [(4^2)-(3√8)] = ___

81 / 14 Signup and view all the answers

Simplify: [(-2)/((x^2)-1)] + [1/(x+1)] - [1/(x-1)] = ___

-4 / [(x^2)-1] Signup and view all the answers

Simplify: [(a^4)(b^3)] / [(a^3)(b^3)] = ___

a Signup and view all the answers

Simplify: (a^b)^c = ___

a^(bc) Signup and view all the answers

What is the highest exponent of x in the polynomial?

The degree of the polynomial Signup and view all the answers

Constants are degree ___ polynomials.

0 Signup and view all the answers

What is the format for first degree polynomials?

y = ax + b Signup and view all the answers

What does the graph of a first degree polynomial look like?

A straight line Signup and view all the answers

What is the more common name for a second degree polynomial?

Quadratic Signup and view all the answers

What is the format for second degree polynomials?

ax^2 + bx + c = 0 Signup and view all the answers

What does the graph of a second degree polynomial look like?

A parabola Signup and view all the answers

What are the steps to graphing polynomials?

<ol> <li>Plug in ± values for x (start with 0 and go outward from there) and solve for y; 2) Plot these points on the graph</li> </ol> Signup and view all the answers

What is another way of saying, 'Find the zeroes of the polynomial'?

Find the roots of the polynomial Signup and view all the answers

What are the ways you can find the roots of quadratics / second degree polynomials?

Factoring or Quadratic Formula Signup and view all the answers

When factoring, the numbers must to form the middle term and to form the last term of a quadratic.

add, multiply Signup and view all the answers

Write the quadratic formula.

x = [(-b) ± √((b^2)-(4ac))] / (2a) Signup and view all the answers

Factor: x^2 - 6x + 9 = ___

(x-3)(x-3) Signup and view all the answers

On the CLEP exam, if asked to factor a problem, what is a shortcut I can do?

Expand the multiple choice answers until I get the polynomial in the question Signup and view all the answers

Factor: x^2 + 6x + 8 = ___

(x+4)(x+2) Signup and view all the answers

Find the roots of the following equation: x^2 + 7x + 9 = ___

x = [(-7) ± √((7^2)-(4⋅1⋅9)] / (2⋅1) Signup and view all the answers

Factor: x^2 - 4 = ___

(x+2)(x-2) Signup and view all the answers

What are the steps to factor higher-order polynomials?

<ol> <li>Factor out any common factors, 2) Guess and check until you factor the problem correctly</li> </ol> Signup and view all the answers

Factor: x^3 - x = ___

x(x+1)(x-1) Signup and view all the answers

How do you expand polynomials?

FOIL Signup and view all the answers

Expand: (x-7)(2x+3) = ___

2x^2 - 11x - 21 Signup and view all the answers

Expand: (x^2 - 7x + 1)(x^2 + 2x + 2) = ___

x^4 - 5x^3 - 4x^2 - 14x + 2 Signup and view all the answers

Study Notes

Adding and Subtracting Fractions

A common denominator is required to add or subtract fractions.

Multiplying and Dividing Fractions

To multiply fractions, multiply the numerators and denominators directly.
When dividing by a fraction, multiply by its reciprocal.

Fraction Computation Examples

Example: (1/3) + (7/5) computes to 26/15 after finding a common denominator.
Another example: [(3x-1) / (x^2-2)] + [1/(x-1)] simplifies to (3x^2 - 4x + 2) / (x^3 - x^2 - 2x + 2).

Properties of Exponents

The square root of x can be expressed as x^(1/2).
(√x)^2 simplifies to x, illustrating that roots and powers cancel each other out.
To multiply terms with the same base, add the exponents.
For terms with different bases but the same exponents, multiply the bases, then raise to the exponent.
When encountering an exponent raised to another exponent, multiply the exponents.
To divide terms with exponents, subtract the exponents.

Simplifying Expressions

Example: Simplifying [(4√9)^8] / [(4^2)-(3√8)] results in 81/14.
To simplify [(-2)/((x^2)-1)] + [1/(x+1)] - [1/(x-1)], the result is -4 / (x^2 - 1).
Example: Simplifying [(a^4)(b^3)] / [(a^3)(b^3)] yields a.
(a^b)^c simplifies to a^(bc).

Polynomial Degree and Formulas

The degree of a polynomial is determined by its highest exponent.
A constant is a polynomial of degree 0.
First-degree polynomial format is y = ax + b; it graphically represents a straight line.
A second-degree polynomial, known as a quadratic, is represented by ax^2 + bx + c = 0.

Graphing Polynomials

To graph polynomials, substitute values for x to find corresponding y values and plot points.

Finding Polynomial Roots

Finding roots of a polynomial is equivalent to finding its zeroes.
Roots of quadratics can be found through factoring or using the quadratic formula.
When factoring, the numbers must add to the middle term's coefficient and multiply to the last term's product.

Quadratic Formula

The quadratic formula allows the computation of roots: x = [(-b) ± √(b^2 - 4ac)] / (2a).

Factoring Examples

Example: x^2 - 6x + 9 factors to (x-3)^2.
x^2 + 6x + 8 factors to (x+4)(x+2).
Roots for x^2 + 7x + 9 are derived from applying the quadratic formula.

Common Factoring Techniques

Common factors should be factored out first in higher-order polynomials.
Example: x^3 - x factors to x(x+1)(x-1).

Polynomial Expansion

To expand polynomials, use the FOIL method for binomials.
Example: Expanding (x-7)(2x+3) yields 2x^2 - 11x - 21.
Example: Expanding (x^2 - 7x + 1)(x^2 + 2x + 2) involves multiplying each term and collecting like terms.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your knowledge of algebra with these flashcards designed for the CLEP exam. Cover essential concepts such as adding and subtracting fractions, multiplying, and dividing by fractions. Perfect for quick review before your test.