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Advanced Algebra 2 - Unit 8 Math Notes 3.1.3 - Vocabulary for Expressions A mathematical expression is a combination of numbers, variables, and operation symbols. Addition and subtraction separate expressions into parts called terms. For example, 4x2 − 3x + 6 is an expression. It has three terms: 4x2, 3x, and 6. The coefficients of the terms with variables are 4 and −3. 6 is called a constant term. A single-variable polynomial is an expression that involves, at most, the operations of addition, subtraction, and multiplication. Most of the polynomials you will work with can be written as expressions with terms of the following form: (any real number)x(whole number) For example, 4x2 – 3x1+ 6x0 is a polynomial, as is the simplified form, 4x2 – 3x+ 6. Also, since 6x0 = 6, 6 itself is a polynomial. 1 The function f(x) = 7x5+ 2.5x3− x + 7 is a polynomial function. 2 A binomial is a polynomial with only two terms, for example, x3− 0.5x and 2x + 5. The following expressions are not polynomials: 2x − 3, polynomials is a rational expression. For example, 1 1 𝑥 2 −2 𝑥 2 −2 , and √𝑥 − 2 . An expression that can be written as the quotient of two , is a rational expression. 8.1.1 - Polynomials, Degree, Coefficients Refer to the Math Notes box in Lesson 3.1.3 for an explanation of a polynomial in one variable. Polynomials with one variable (often x) are usually arranged with powers of x in order, starting with the highest, left to right. The highest power of the variable in a polynomial of one variable is called the degree of the polynomial. The numbers that multiply each term are called coefficients. See the examples below. 1 1 Example 1: f(x) = 7x5 + 2.5x3 − 𝑥 + 7 is a polynomial function of degree 5 with coefficients 7, 0, 2.5, 0, − , and 7. Note that the last 2 2 term, 7, is called the constant term but represents the variable expression 7x0, since x0 = 1. Example 2: y = 2(x + 2)(x + 5) is a polynomial in factored form with degree 2 because it can be written in standard form as y = 2x2 + 14x + 20. It has coefficients 2, 14, and 20. 8.1.2 - Roots and Zeros The roots of a polynomial function, p(x), are the solutions of the equation p(x) = 0. Another name for the roots of a function is zeros of a function because at each root, the value of the function is zero. The real roots (or zeros) of a function have the same value as the x-values of the x-intercepts of its graph because the x-intercepts are the points where the y-value of the function is zero. Sometimes roots can be found by factoring and solving for p(x) = 0. In the Parabola Lab investigation (Lesson 2.1.2), you discovered how to make a parabola “sit” on the x-axis (the polynomial has one root), and you looked at ways of making parabolas intersect the x-axis in two specific places (two roots). 8.1.3 - Notation for Polynomials The general equation of a second-degree (quadratic) polynomial is often written in the form f(x) = ax2 + bx + c, and the general equation of a third-degree (cubic) polynomial is often written in the formf(x) = ax3 + bx2 + cx + d. For a polynomial with an undetermined degree n, it is unknown how many letters will be needed for the coefficients. Instead of using a, b, c, d, e, etc., mathematicians use only the letter a, and they used subscripts, as shown below. f(x) = (an)xn + (an−1)x(n−1) + … + (a1)x1 + a0 This general polynomial has degree n and coefficients an , an−1, …, a1, a0. For example, for 7x4 − 5x3 + 3x2 + 7x + 8, the degree is 4. In this specific case, an is a4 and a4 = 7, an−1 is a3 = −5, an−2 is a2 = 3, a1 = 7, and a0 = 8. 8.2.1 - Imaginary and Complex Numbers The imaginary number i is defined as the square root of –1, so i = √−1. Therefore i2 = –1, and the two solutions of the equation x2 + 1 = 0 are x = i and x = –i. In general, i follows the rules of real number arithmetic. The sum of two imaginary numbers is imaginary (unless it is 0). Multiplying the imaginary number i by every possible real number would yield the set of all the imaginary numbers. The set of numbers that solve equations of the form x2 = (a negative real number) is called the set of imaginary numbers. Imaginary numbers are not positive, negative, or zero. The collection (set) of positive and negative numbers (integers, rational numbers (fractions), and irrational numbers), are referred to as the real numbers. The sum of a real number (other than zero) and an imaginary number, such as 2 + i, is generally neither real nor imaginary. Numbers such as these, which can be written in the form a + bi, where a and bare real numbers, are called complex numbers. Each complex number has a real component, a, and an imaginary component, bi. The real numbers are considered to be complex numbers with b = 0, and the imaginary numbers are complex numbers with a = 0. 8.2.2 - The Discriminant and Complex Conjugates With the introduction of complex numbers, the use of the terms roots and zeros of polynomials expands to include complex numbers that are solutions of the equations when p(x) = 0. For any quadratic equation ax2 + bx + c = 0, you can determine whether the roots are real or complex by examining the part of the quadratic formula that is under the square-root sign. The value of b2− 4ac is known as the discriminant. The roots are real when b2 − 4ac ≥ 0 and complex when b2 − 4ac < 0. For example, in the equation 2x2 − 3x + 5 = 0, b2 − 4ac = (−3)2− 4(2)(5) = −31 < 0, so the equation has two complex roots and the parabola y = 2x2 − 3x + 5 does not intersect the x-axis. Complex roots of quadratic equations with real coefficients will have the form a − bi and a + bi, which are called complex conjugates. The sum and product of two complex conjugates are always real numbers. For example, the conjugate for the complex number −5 + 3i, is −5 − 3i. (−5 + 3i) + (−5 − 3i) = −10 and (−5 + 3i)(−5 − 3i) = 25 − 9i2 = 34. 8.2.3 - Graphing Complex Numbers To represent complex numbers, an imaginary axis and a real axis are needed. Real numbers are on the horizontal axis and imaginary numbers are on the vertical axis, as shown in the examples below. This representation is called the complex plane. In the complex plane, a + bi is located at the point (a, b).The number 2 + 3i is located at the point (2, 3). The number i or 0 + 1i is located at (0, 1). The number –2 or –2 + 0iis located at (–2, 0). The absolute value of a complex number is its distance from the origin. To find the absolute value, calculate the distance from (0, 0) to (a, b): |𝑎 + 𝑏𝑖| = √𝑎2 + 𝑏 2 Examples: 8.3.1 - Polynomial Division The examples below show two methods for dividing x4 − 6x3 + 18x − 1 by x − 2. In both cases, the remainder is written as a fraction. Using long division: Using an area model: Answer: Therefore, and 8.3.2 - Polynomial Theorems Factor Theorem: If x = a is a zero of a polynomial function, then p(x) then (x − a) is a factor of the polynomial, and if (x − a) is a factor of the polynomial, then x = a is a zero of the polynomial. Example: If x = −3, x = 2, and x = 1 ± 𝑖√7 are zeros of the polynomial p(x) = 2x4 − x3 + 20x − 48. According to the Factor Theorem, (x + 3),(x − 2), (𝑥 − (1 + 𝑖√7)), and (𝑥 − (1 − 𝑖√7)), are factors of the polynomial. Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra states that every one-variable polynomial has at least one complex root (remember that every real number is also a complex number). This theorem can be used to show that every polynomial of degree n has n complex roots. This also means that the polynomial has n linear factors, since for every root a, (x – a) is a linear factor. Note that the total of n roots might include roots that occur multiple times. For example, the third degree polynomial p(x) = x3 − 3x2 + 4 has three roots—a root at x = –1 and a double root at x = 2. The three linear factors are (x + 1), (x – 2), and (x – 2) Integral Zero Theorem: For any polynomial with integral coefficients, if an integer is a zero of the polynomial, it must be a factor of the constant term. Example: Suppose the integers a, b, c, and d are zeros of a polynomial. Then, according to the Factor Theorem, (x − a)(x − b)(x − c)(x − d) are factors of the polynomial.When you multiply these factors together, the constant term will be abcd, so a, b, c, and d are factors of the constant term. Remainder Theorem: For any number c, when a polynomial p(x) is divided by (x – c), the remainder is p(c). For example, if the polynomial p(x) = x4 −x3 + 20x –48 is divided by (x − 5), the remainder is p(5) = 552. Note that it follows from the Remainder Theorem that if p(c)= 0, then (x – c), is a factor. For example, if p(x) = x3 −3x2 + 4, one solution to x3 −3x2 + 4 = 0 is x = –1. Since p(–1) = 0, then (x + 1) is a factor. 8.3.3 - Factoring Sums and Differences The difference of two squares can be factored: a2 − b2 = (a + b)(a − b) The sum of two cubes can be factored: a3 + b3 = (a + b)(a2 − ab + b2) The difference of two cubes can be factored: a3 − b3 = (a − b)(a2 + ab + b2)