2.2 Polynomial Representations

Polynomials are mathematical expressions with variables and exponents. They come in different forms like standard, factored, and expanded. Understanding these forms helps us manipulate and solve polynomial equations more easily.

Arithmetic operations on polynomials include addition, subtraction, multiplication, and division. These operations are crucial for solving complex problems and simplifying expressions. Factoring and evaluation techniques further enhance our ability to analyze and work with polynomials effectively.

Polynomial Representations and Operations

Forms of polynomial representation

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• Standard form represents polynomials as a sum of terms with decreasing powers of the variable $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ (cubic polynomial $3x^3 - 2x^2 + 5x - 1$)
• Factored form expresses polynomials as a product of linear factors $(a_1x + b_1)(a_2x + b_2)...(a_nx + b_n)$ (quadratic $(x - 1)(x + 2)$)
• Expanded form writes polynomials with each term expanded and like terms combined $a_0 + a_1x + a_2x^2 + ... + a_nx^n$ (quartic $x^4 + 3x^3 - 4x^2 + 2x - 1$)

Arithmetic operations on polynomials

• Addition and subtraction combine like terms by adding or subtracting their coefficients $(2x^2 + 3x - 1) + (x^2 - 2x + 4) = 3x^2 + x + 3$
• Multiplication involves multiplying each term of one polynomial by each term of the other polynomial and combining like terms $(2x + 1)(x - 3) = 2x^2 - 5x - 3$
• Division uses long division or synthetic division to divide polynomials and find quotients and remainders $(x^3 + 2x^2 - 3x + 4) \div (x + 1)$ yields a quotient of $x^2 + x - 4$ and a remainder of $0$

Polynomial division techniques

• Long division divides the highest degree term of the dividend by the highest degree term of the divisor and repeats the process with the remainder $(x^3 - 2x^2 + 3x - 4) \div (x - 1)$ yields a quotient of $x^2 - x + 3$ and a remainder of $-1$
• Synthetic division provides a shortcut method for dividing a polynomial by a linear factor $(x - a)$ using the coefficients and the negative of the constant term in the linear factor $(2x^3 + 3x^2 - 5x + 1) \div (x + 2)$ yields a quotient of $2x^2 - x - 2$ and a remainder of $5$

Factoring methods for polynomials

• Grouping involves grouping terms and factoring out common factors $ax^3 + bx^2 + cx + d = (ax^2 + bx) + (cx + d)$ (quadratic $6x^2 + 5x - 4 = (2x - 1)(3x + 4)$)
• Difference of squares factors polynomials in the form $a^2 - b^2 = (a + b)(a - b)$ (binomial $x^2 - 9 = (x + 3)(x - 3)$)
• Sum or difference of cubes factors polynomials in the form $a^3 \pm b^3$ where $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ (cubic $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$)

Evaluation and analysis of polynomials

• Substitution replaces the variable with a given value and simplifies the expression to evaluate polynomials $f(x) = 2x^2 - 3x + 1$, $f(2) = 2(2)^2 - 3(2) + 1 = 3$
• Finding zeros (roots) involves setting the polynomial equal to zero and solving for the variable using factoring, quadratic formula, or other methods $f(x) = x^2 - 5x + 6 = (x - 2)(x - 3)$, zeros at $x = 2$ and $x = 3$
• Graphing polynomials requires plotting points or using transformations to identify key features such as y-intercept, zeros, and end behavior $f(x) = x^2 - 4x + 3$, vertex $(-b/2a, f(-b/2a))$, zeros at $x = 1$ and $x = 3$, y-intercept at $(0, 3)$