Pre-Algebra Unit 10 Review: Polynomials

What Are Polynomials?

• Polynomials are mathematical expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication
• The general form of a polynomial is $a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0$, where $a_n, a_{n-1}, ..., a_2, a_1, a_0$ are constants and $n$ is a non-negative integer
• Polynomials cannot have variables in the denominator or under a square root
• The degree of a polynomial is the highest power of the variable in the expression ($x^3 + 2x + 1$ has a degree of 3)
• Polynomials can have one or more variables ($3x^2y + 2xy^2 - 5$ is a polynomial with two variables, $x$ and $y$)
• The standard form of a polynomial arranges the terms in descending order of the degree ($2x^3 - 4x^2 + 3x - 1$)
• Polynomials are closed under addition, subtraction, and multiplication, meaning the result of these operations on polynomials is always another polynomial

Parts of a Polynomial

• Terms are the individual parts of a polynomial separated by addition or subtraction ($3x^2$, $-4x$, and $5$ are the terms in $3x^2 - 4x + 5$)
  • Like terms have the same variables raised to the same powers ($2x^2$ and $-3x^2$ are like terms)
• Coefficients are the numerical factors of each term ($3$ is the coefficient in $3x^2$)
  • The leading coefficient is the coefficient of the term with the highest degree ($2$ in $2x^3 - 4x^2 + 3x - 1$)
• Variables are the letters in a polynomial that represent unknown values ($x$ and $y$ in $3x^2y + 2xy^2 - 5$)
• Exponents indicate the power to which a variable is raised ($2$ in $x^2$)
• Constants are the terms without variables ($5$ in $3x^2 - 4x + 5$)
• The degree of a term is the sum of the exponents of its variables ($3x^2y$ has a degree of 3)
  • The degree of a constant term is 0

Types of Polynomials

• Monomials are polynomials with only one term ($3x^2$, $-4y$, $5$)
• Binomials are polynomials with exactly two terms ($2x + 3$, $x^2 - 4$)
• Trinomials are polynomials with exactly three terms ($x^2 + 2x + 1$, $3x^2 - 4x + 5$)
• Quadratic polynomials have a degree of 2 ($ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$)
• Cubic polynomials have a degree of 3 ($ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants and $a \neq 0$)
• Quartic polynomials have a degree of 4 ($ax^4 + bx^3 + cx^2 + dx + e$, where $a$, $b$, $c$, $d$, and $e$ are constants and $a \neq 0$)
• The zero polynomial is a polynomial where all coefficients are 0 ($0x^2 + 0x + 0$)

Adding and Subtracting Polynomials

• To add or subtract polynomials, combine like terms by adding or subtracting their coefficients
  • $(2x^2 + 3x - 1) + (4x^2 - 2x + 5) = 6x^2 + x + 4$
  • $(5x^3 - 2x + 3) - (3x^3 + 4x - 1) = 2x^3 - 6x + 4$
• Arrange the terms of the resulting polynomial in descending order of the degree
• When subtracting polynomials, distribute the negative sign to each term in the second polynomial before combining like terms
  • $(4x^2 - 3x + 2) - (2x^2 + x - 3) = 4x^2 - 3x + 2 - 2x^2 - x + 3 = 2x^2 - 4x + 5$
• The degree of the sum or difference of two polynomials is the greater of the degrees of the two polynomials
• Adding or subtracting polynomials with different variables requires keeping the terms separate ($2x^2 + 3y - 1$ cannot be simplified further)

Multiplying Polynomials

• To multiply polynomials, use the distributive property to multiply each term of one polynomial by every term of the other polynomial
  • $(2x + 3)(x - 1) = 2x^2 - 2x + 3x - 3 = 2x^2 + x - 3$
• Combine like terms in the resulting polynomial and arrange them in descending order of the degree
• The degree of the product of two polynomials is the sum of the degrees of the two polynomials
• Special products:
  • Square of a binomial: $(a + b)^2 = a^2 + 2ab + b^2$
  • Difference of squares: $(a^2 - b^2) = (a + b)(a - b)$
  • Sum and difference of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
• Multiplying polynomials with different variables results in a polynomial with all the variables present ($(2x + 3)(y - 1) = 2xy - 2x + 3y - 3$)

Dividing Polynomials

• Long division is a method for dividing polynomials by arranging the terms in descending order and dividing the leading term of the dividend by the leading term of the divisor
  • Multiply the quotient by the divisor, subtract the result from the dividend, and bring down the next term
  • Repeat the process until the degree of the remainder is less than the degree of the divisor
• Synthetic division is a shortcut for dividing a polynomial by a linear factor $(x - a)$
  • Write the coefficients of the polynomial in descending order, with missing terms as 0
  • Bring down the leading coefficient, multiply it by $a$, add the result to the next coefficient, and repeat until the last coefficient
  • The last number is the remainder, and the other numbers form the quotient
• The quotient of two polynomials may not always be a polynomial (e.g., $(x^2 + 1) \div (x - 1) = x + 1 + \frac{1}{x - 1}$)
• The remainder theorem states that the remainder of a polynomial $P(x)$ divided by $(x - a)$ is equal to $P(a)$

Factoring Polynomials

• Factoring is the process of expressing a polynomial as a product of its factors
• Common factors can be factored out of a polynomial by finding the greatest common factor (GCF) of all terms ($6x^2 + 9x = 3x(2x + 3)$)
• Factoring by grouping involves grouping terms, factoring out common factors, and then factoring the resulting polynomial ($ax^3 + bx^2 + cx + d = (ax^2 + bx) + (cx + d) = x(ax + b) + (cx + d)$)
• Trinomials of the form $ax^2 + bx + c$ can be factored by finding two numbers whose sum is $b$ and whose product is $ac$ ($2x^2 + 7x + 3 = (2x + 1)(x + 3)$)
• Perfect square trinomials can be factored as the square of a binomial ($x^2 + 6x + 9 = (x + 3)^2$)
• The difference of squares can be factored as the product of the sum and difference of the square roots ($x^2 - 9 = (x + 3)(x - 3)$)
• Cubic polynomials can be factored by grouping or using the sum and difference of cubes formulas

Solving Polynomial Equations

• To solve a polynomial equation, set the polynomial equal to zero and find the values of the variable that make the equation true
• For linear equations, isolate the variable on one side of the equation ($2x - 3 = 7$ becomes $2x = 10$, so $x = 5$)
• For quadratic equations, factor the polynomial and apply the zero product property ($x^2 - 5x + 6 = 0$ factors to $(x - 2)(x - 3) = 0$, so $x = 2$ or $x = 3$)
  • The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ can be used to solve any quadratic equation $ax^2 + bx + c = 0$
• For higher-degree polynomials, factor the polynomial if possible and apply the zero product property
  • Rational root theorem can help find potential rational roots of a polynomial equation
  • Descartes' rule of signs can determine the number of positive and negative real roots of a polynomial equation
• Graphing a polynomial can help identify the approximate locations of its roots (x-intercepts)

Real-World Applications

• Modeling the height of a projectile over time with a quadratic function ($h(t) = -4.9t^2 + v_0t + h_0$, where $h$ is height, $t$ is time, $v_0$ is initial velocity, and $h_0$ is initial height)
• Describing the volume of a rectangular box with a polynomial ($V(x) = x(10 - 2x)(8 - 2x)$, where $x$ is the side length and $V$ is the volume)
• Calculating the area of a rectangle with a fixed perimeter using a quadratic function ($A(x) = 20x - 2x^2$, where $x$ is one side length and $A$ is the area)
• Modeling population growth or decay with polynomial functions ($P(t) = 100(1 + 0.05)^t$, where $P$ is population, $t$ is time, and the growth rate is 5%)
• Representing the cost, revenue, and profit of a business with polynomial functions ($C(x) = 500 + 2x$, $R(x) = 5x$, $P(x) = R(x) - C(x)$, where $x$ is the number of units)
• Describing the motion of a spring or pendulum with polynomial functions ($F(x) = -kx$, where $F$ is the restoring force, $k$ is the spring constant, and $x$ is the displacement)