D1

Polynomial Functions

Definition and Key Terms

A polynomial function of degree n is defined by:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

• Degree: The highest power of x in the polynomial.

• Leading coefficient: The coefficient of the term with the highest degree.

• Leading term: The term with the highest degree.

• Constant term: The term without x (i.e., $a_0$).

Example: For $f(x) = 6x^4 - 2x^3 + 5x - 7$:

• Degree: 4

• Leading coefficient: 6

• Leading term: $6x^4$

• Constant term: -7

Non-Polynomial Functions

• Functions with negative or fractional exponents, or variables in the exponent, are not polynomials.

• Example: $f(x) = x^{-4}$ is not a polynomial (negative exponent).

• Example: $g(x) = 2^x$ is not a polynomial (variable in exponent).

Classification of Polynomials

Degree and Number of Terms

Polynomial	Degree	Number of Terms	Name/Classification
$f(x) = 8x^2$	2	1	Quadratic monomial
$f(x) = x^2 - 4$	2	2	Quadratic binomial
$f(x) = 3x^3 + 7$	3	2	Cubic binomial
$f(x) = 6x^4 + x^2 - 7$	4	3	Quartic trinomial

Operations on Polynomials

Addition and Subtraction

Polynomials are added or subtracted by combining like terms.

• Example: $f(x) = 3x^2 + 7$, $g(x) = 5x^2 - 13$

• $f(x) + g(x) = (3x^2 + 7) + (5x^2 - 13) = 8x^2 - 6$

• $f(x) - g(x) = (3x^2 + 7) - (5x^2 - 13) = -2x^2 + 20$

Multiplication (Distributive Property)

The distributive property $a(b + c) = ab + ac$ is used to multiply polynomials. For binomials, the FOIL method (First, Outer, Inner, Last) is often used.

• Example: $-3(x^2 + 4x - 2)$

• $= -3x^2 - 12x + 6$

• Example: $(x - 4)(2x + 9)$

• $= x \cdot 2x + x \cdot 9 - 4 \cdot 2x - 4 \cdot 9 = 2x^2 + 9x - 8x - 36 = 2x^2 + x - 36$

Division of Polynomials

Polynomials can be divided by monomials or other polynomials. The division algorithm states:

$f(x) = d(x)q(x) + r(x)$

• $d(x)$: divisor

• $q(x)$: quotient

• $r(x)$: remainder (degree less than $d(x)$)

Example: Divide $18 - 24x + 42x^2$ by $6x$:

• $\frac{18}{6x} = 3x^{-1}$

• $\frac{-24x}{6x} = -4$

• $\frac{42x^2}{6x} = 7x$

• Quotient: $3x^{-1} - 4 + 7x$

Long Division of Polynomials

To divide polynomials, arrange terms in descending order and divide step by step, subtracting and bringing down terms as needed.

• Example: $2x^2 - 7x - 15$ divided by $x + 2$

• Quotient: $2x - 11$, Remainder: $-13$

• Expressed as: $2x - 11 + \frac{-13}{x+2}$

Factoring Polynomials

Factoring Techniques

• Look for the greatest common factor (GCF) first.

• Apply the distributive property to factor expressions.

• Identify the number of terms: binomial, trinomial, etc.

• For binomials, check for difference of squares: $a^2 - b^2 = (a - b)(a + b)$

• For trinomials, use factoring formulas or trial and error.

• Group terms for polynomials with more than three terms.

• Check your answer by multiplying all factors.

Examples of Factoring

• $3x^2 - 21x^2 - 24x = 3x(x^2 - 8x - 8)$

• $2x^2 - 5x - 12 = (2x + 3)(x - 4)$

• $x^2 + 9$ cannot be factored over the real numbers.

Factoring by Division

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

• Use division to find other linear factors.

• Example: Given $x + 3$ is a factor of $f(x) = x^3 + 2x^2 - 5x^2 - 35x$, divide to find other factors.

Practice Problems

• Add, subtract, multiply, and divide polynomials as shown in the examples above.

• Factor polynomials using the strategies provided.

• Write polynomials as products of linear factors when possible.

Summary of Factoring Strategies

• Always look for the greatest common factor (GCF) first.

• Apply the distributive property.

• Identify the number of terms and use appropriate factoring methods.

• Check for special patterns (difference of squares, perfect square trinomials).

• Group terms if necessary.

• Check your answer by multiplying all factors.