The final section of our PERT Math Study Guide is about polynomials. Here, we will define all relevant terms and break down polynomial operations.
Definition of Polynomials
A polynomial is an algebraic expression composed of terms. These terms fall into two categories: indeterminates (or variables), including variables raised to non-negative integer powers, and coefficients.
Polynomials can have multiple terms and are generally written in descending order of powers.
Example: $3x^2 -4x + 7$ is a polynomial with three terms: $3x^2, -4x,$ and $7.$
Polynomial Terminology
Degree: The highest power of the variable. For instance, in $4x^3 + 2x -1,$ the degree is $3$.
Leading Coefficient: The coefficient of the term with the highest degree. In $5x^2 -3x + 8,$ the leading coefficient is $5$.
Constant Term: A term with no variable. In the polynomial $5x^2 -3x + 8$, $8$ is a constant term.
Operations with Polynomials
Addition/Subtraction: Combine like terms (terms with the same variable and power). For example:
$(3x^2 + 4x + 5) $ $ + (2x^2 -3x -1) $ $ = 5x^2 + x + 4$
Multiplication: Distribute each term in one polynomial by each term in the other. Use the FOIL method for binomials or the distributive property for larger polynomials. For example:
$(x + 2)(x -3) = x^2 -x -6$
Division: Divide a polynomial by a monomial or another polynomial, often using long division or synthetic division.
Factoring Polynomials
Greatest Common Factor (GCF): Factor out the largest term common to all terms. For example:
$6x^3 + 9x^2 $ $ = 3x^2 (2x + 3)$
Factoring Trinomials: For trinomials in $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$.
Special Factoring Patterns:
- Difference of Squares: $a^2 + b^2 = (a + b)(a – b)$
- Perfect Square Trinomials: $a^2 ± 2ab + b^2 $ $ = (a ± b)^2$
Polynomial Functions
A polynomial function is defined by a polynomial equation, such as $f(x) = 2x^3 -x^2 + 4x -5$. Polynomial functions are continuous with no breaks, holes, or gaps.
Roots/Zeros: The roots (or zeros) of a polynomial function are values of $x$ that make $f(x) = 0$. These can be found by factoring the polynomial and setting each factor to zero.
Key Tip: Arrange polynomial terms in descending order of their exponents to make the expression clearer and to simplify future calculations. This helps you easily identify the degree and leading term of the polynomial.