BackOperations with Polynomials and Rational Expressions
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Polynomial Operations and Factoring
Addition and Subtraction of Polynomials
Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. Adding or subtracting polynomials involves combining like terms—terms with the same variable raised to the same power.
Adding Polynomials: Align like terms and add their coefficients.
Subtracting Polynomials: Distribute the negative sign, then combine like terms.
Example: Add Combine like terms:
Multiplying Polynomials
Multiplication of polynomials can involve a monomial (single term) or two general polynomials. The distributive property is used to multiply each term in one polynomial by each term in the other.
Multiplying a Monomial and a Polynomial: Multiply the monomial by each term in the polynomial. Example:
Multiplying Two Polynomials: Use the distributive property (also known as the FOIL method for binomials). Example:
Squares of Binomials
The square of a binomial is a special product that follows a specific pattern:
Formula:
Example:
Dividing a Polynomial by a Monomial
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
Example:
Factoring Trinomials
Factoring is the process of writing a polynomial as a product of its factors. For trinomials of the form , find two numbers that multiply to and add to .
Example: Factor Find numbers that multiply to 6 and add to 5: 2 and 3. So,
Factoring Perfect Square Trinomials
A perfect square trinomial is the result of squaring a binomial. It can be factored using the reverse of the square of a binomial formula.
Formula:
Example:
Operations with Rational Expressions
Multiplying and Dividing Rational Expressions
Rational expressions are fractions where the numerator and denominator are polynomials. To multiply, multiply numerators and denominators; to divide, multiply by the reciprocal of the divisor.
Multiplying:
Dividing:
Example: (after canceling )
Adding and Subtracting Rational Expressions with the Same Denominator
When rational expressions have the same denominator, add or subtract the numerators and keep the denominator unchanged.
Formula:
Example:
Simplifying Square Roots
Product Rule for Square Roots
The product rule for square roots states that the square root of a product is the product of the square roots, provided all expressions are non-negative.
Formula:
Example:
