Algebra

The One Thing

Polynomial and rational algebra is solved by rewriting expressions systematically: combine like terms, factor where possible, track excluded values, and apply equivalent transformations. Factored form reveals zeros and supports equation solving, while expanded form supports combination, degree identification, and coefficient analysis.

Definitions and Results

• Polynomial: An expression made from constants and variables with nonnegative integer exponents, such as $3x^2-5x+1$.
• Term: A number, variable, or product of numbers and variables separated by addition or subtraction signs.
• Coefficient: The numerical factor multiplying a variable, such as $4$ in $4x^3$.
• Degree: The greatest exponent of the variable in a polynomial, or the greatest sum of exponents in any term for a polynomial with several variables.
• Like Terms: Terms with the same variables raised to the same powers; only their coefficients can be combined.
• Standard Form: A polynomial written in descending order of degree, such as $2x^3-x^2+4x-7$.
• Factoring: Rewriting an expression as a product of simpler expressions, such as

$$x^2-9=(x-3)(x+3).$$

• Greatest Common Factor: The largest factor shared by every term in an expression.
• Difference of Squares: The identity

$$a^2-b^2=(a-b)(a+b).$$

• Perfect-Square Formulas:

$$a^2+2ab+b^2=(a+b{)}^2,$$ $$a^2-2ab+b^2=(a-b{)}^2.$$

• Rational Expression: An expression written as the quotient of two polynomials, such as

$$\frac{x+2}{x-5}.$$

• Excluded Value: A value that makes the denominator of a rational expression zero and is therefore not allowed.
• Equivalent Expressions: Expressions that have the same value for every allowed value of the variable.
• Polynomial Equation: An equation in which one or both sides contain polynomial expressions.
• Quadratic Equation: An equation that can be written as

$$a{x}^2+bx+c=0,$$ where $a\ne 0$.

• Zero Product Property: If $ab=0$, then $a=0$ or $b=0$.
• Root or Solution: A value that makes an equation true or makes a function equal to zero.
• Multiplicity: The number of times a factor occurs in a factored polynomial; it can affect whether a graph meets or crosses the $x$-axis.
• Remainder Theorem: When $p(x)$ is divided by $x-r$, the remainder is $p(r)$.
• Factor Theorem: If $p(r)=0$, then $x-r$ is a factor of $p(x)$, and conversely.
• Polynomial Division Relationship:

$$\text{dividend}=\text{divisor}\times \text{quotient}+\text{remainder}.$$

• Quadratic Formula: For $a{x}^2+bx+c=0$,

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

• Discriminant: The quantity $b^2-4ac$. If it is positive, there are two real solutions; if zero, there is one repeated real solution; if negative, there are no real solutions.
• Fundamental Theorem of Algebra: A degree-$n$ polynomial has exactly $n$ complex roots when counted with multiplicity.
• Graphical Results: A polynomial function is continuous. Its end behavior is determined mainly by its degree and leading coefficient. Its zeros are the $x$-values where its graph intersects or touches the $x$-axis.
• Rational Root Theorem: For a polynomial with integer coefficients, possible rational zeros are factors of the constant term divided by factors of the leading coefficient.
• Equivalent Transformations: Algebraic transformations preserve equation solutions when each step is reversible and all restrictions are respected.

Worked Methods

1. Rewriting and Combining Polynomials

• Identify the terms and their coefficients.
• Arrange the polynomial in standard form if required.
• Combine only like terms.
• For addition or subtraction, distribute a negative sign across every term when necessary.

The distributive property is $$a(b+c)=ab+ac.$$

Expanded form is useful for combining terms and identifying degree and coefficients. Factored form is useful for finding zeros.

2. Multiplying Polynomials

• Multiply every term in the first polynomial by every term in the second.
• Use the distributive property for each multiplication.
• Combine like terms.
• Write the result in standard form.

The essential requirement is that every term in one polynomial is multiplied by every term in the other polynomial.

3. Factoring Polynomials

• First identify and factor out the greatest common factor.
• Examine the remaining expression for a recognizable pattern.
• Apply trinomial or perfect-square patterns where appropriate.
• Apply the difference-of-squares identity when the expression has the form $a^2-b^2$.
• Check the factorization by multiplying the factors.

The named example is $$x^2-9=(x-3)(x+3).$$

The relevant identities are $$a^2-b^2=(a-b)(a+b),$$ $$a^2+2ab+b^2=(a+b{)}^2,$$ and $$a^2-2ab+b^2=(a-b{)}^2.$$

Factoring is structurally important because it exposes the factors that determine polynomial zeros.

4. Simplifying Rational Expressions

• Factor the numerator and denominator completely.
• Identify common factors.
• Cancel only common factors, not individual terms separated by addition or subtraction.
• Record values that make the original denominator zero.
• Retain those excluded values even if the expression simplifies to a shorter form.

For example, in $$\frac{x+2}{x-5},$$ the value $x=5$ is excluded because it makes the denominator zero.

A simplified rational expression remains subject to every restriction imposed by the original denominator.

5. Operating with Rational Expressions

Addition and Subtraction

• Factor denominators where useful.
• Determine a common denominator.
• Rewrite each rational expression using that denominator.
• Add or subtract the numerators.
• Simplify the resulting expression.
• Retain all excluded values from the original denominators.

Multiplication and Division

• Factor all numerators and denominators.
• For multiplication, multiply numerators and denominators, canceling common factors where permitted.
• For division, multiply by the reciprocal of the second rational expression.
• State restrictions from the original denominators and, for division, restrictions ensuring the divisor is not zero.
• Simplify and check the result.

6. Polynomial Division, the Remainder Theorem, and the Factor Theorem

For polynomial division:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the divisor by the resulting quotient term.
• Subtract.
• Repeat until the remainder has lower degree than the divisor.
• Express the result using

$$\text{dividend}=\text{divisor}\times \text{quotient}+\text{remainder}.$$

If the divisor is $x-r$, use the Remainder Theorem: the remainder is $p(r)$.

Then apply the Factor Theorem:

• If $p(r)=0$, $x-r$ is a factor of $p(x)$.
• If $x-r$ is a factor of $p(x)$, then $p(r)=0$.

The Rational Root Theorem can be used before testing values. For integer-coefficient polynomials, list possible rational zeros as factors of the constant term divided by factors of the leading coefficient.

7. Solving Polynomial Equations by Factoring

• Move all terms to one side.
• Write the equation in standard form with zero on the other side.
• Factor the polynomial completely.
• Apply the Zero Product Property.
• Set each factor equal to zero.
• Solve the resulting equations.
• Check the solutions in the original equation where restrictions may be present.

The key principle is $$ab=0⟹a=0\text{ or }b=0.$$

Roots may have multiplicity when the same factor occurs more than once. Multiplicity can affect whether the graph crosses or merely touches the $x$-axis.

8. Solving Quadratic Equations

By Factoring

• Write the equation as $a{x}^2+bx+c=0$.
• Factor the quadratic.
• Apply the Zero Product Property.
• Solve each linear equation.
• Check the solutions.

By Completing the Square

• Move the constant term to the other side.
• If necessary, divide by the coefficient of $x^2$.
• Add the appropriate quantity to both sides to form a perfect square.
• Rewrite the quadratic in vertex form.
• Take square roots of both sides.
• Solve for $x$.

Completing the square can transform a quadratic equation into vertex form and can also be used to derive or apply the quadratic formula.

By the Quadratic Formula

• Put the equation into the form

$$a{x}^2+bx+c=0.$$

• Identify $a$, $b$, and $c$.
• Substitute them into

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

• Evaluate the discriminant $b^2-4ac$.
• Simplify both possible values of $x$.
• Check the solutions in the original equation.

Factoring, completing the square, graphing, and the quadratic formula describe the same solutions in different forms.

9. Interpreting Polynomial Graphs

• Find the zeros by solving $p(x)=0$.
• Use factor multiplicity to determine whether the graph crosses or touches the $x$-axis.
• Use the degree and leading coefficient to determine end behavior.
• Use continuity to interpret the behavior between points.

A polynomial’s zeros are the $x$-values where its graph intersects or touches the $x$-axis.

10. Verifying Algebraic Identities

An identity can be verified by:

• Expanding both sides.
• Factoring one or both sides.
• Substituting appropriate values.
• Comparing the two sides after simplification.

An identity must hold for every allowed value of the variable, unlike an equation that may hold only for particular solutions.

Where It Goes Wrong

• Combining unlike terms instead of only like terms; terms must have the same variables raised to the same powers.
• Failing to multiply every term in one polynomial by every term in the other when using the distributive property.
• Canceling terms rather than common factors in rational expressions.
• Omitting values that make the original denominator zero after simplifying a rational expression.
• Applying the Zero Product Property before first writing a polynomial equation in factored form with zero on one side.
• Losing restrictions or accepting extraneous solutions when multiplying by expressions containing variables; proposed solutions must be checked in the original equation.

What Gets Asked

• Identify the terms, coefficients, degree, like terms, and standard form of a polynomial.
• Add, subtract, multiply, and rewrite polynomial expressions.
• Factor expressions using a greatest common factor, trinomials, perfect-square formulas, or the difference of squares.
• Simplify, multiply, divide, add, and subtract rational expressions while stating excluded values.
• Perform polynomial division and use the Remainder Theorem or Factor Theorem.
• List possible rational zeros using the Rational Root Theorem.
• Solve polynomial equations by factoring and the Zero Product Property.
• Solve quadratic equations by factoring, completing the square, and the quadratic formula.
• Interpret the discriminant and determine the number of real solutions.
• Identify roots, zeros, and multiplicities and relate them to polynomial graphs.
• Determine graph end behavior from degree and leading coefficient.
• Verify algebraic identities by expanding, factoring, substituting, or comparing both sides.
• Translate among symbolic expressions, graphs, tables, and real-world situations.
• Check proposed solutions in the original equation, especially for rational equations and transformations involving variable expressions.