3. POLYNOMIALS

## Introduction Welcome to this lesson on Polynomials! Polynomials are fundamental algebraic expressions that you've likely encountered in various forms, such as linear equations ( \(y = mx + c\) ) and quadratic equations ( \(ax^2 + bx + c = 0\) ). In Further Mathematics, we delve deeper into their properties, operations, and applications. Polynomials are widely used in science, engineering, economics, and computer graphics to model a vast range of phenomena, from the trajectory of a projectile to the design of complex curves. Understanding them is crucial for advanced mathematical study. In this lesson, we will define what a polynomial is, explore different types, learn how to perform operations with them, and discover powerful theorems that help us understand their roots and factors. ## Key Concepts ### Definition of a Polynomial A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. #### General Form A polynomial in a single variable \(x\) can be written in the general form: \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \] Where: * \(a_n, a_{n-1}, \dots, a_1, a_0\) are the **coefficients** (real or complex numbers). * \(x\) is the **variable**. * \(n\) is a **non-negative integer**, representing the highest power of \(x\). This is known as the **degree** of the polynomial, provided that \(a_n \neq 0\). * \(a_n x^n\) is the **leading term**, and \(a_n\) is the **leading coefficient**. * \(a_0\) is the **constant term**. **Example:** \(P(x) = 3x^4 - 2x^2 + 5x - 7\) * Degree: 4 * Leading coefficient: 3 * Coefficients: 3, 0 (for \(x^3\)), -2, 5, -7 * Constant term: -7 ### Types of Polynomials Polynomials can be classified by their degree or by the number of terms. #### By Degree: * **Degree 0 (Constant Polynomial):** \(P(x) = a_0\) (e.g., \(P(x) = 5\)) * **Degree 1 (Linear Polynomial):** \(P(x) = a_1 x + a_0\) (e.g., \(P(x) = 2x - 3\)) * **Degree 2 (Quadratic Polynomial):** \(P(x) = a_2 x^2 + a_1 x + a_0\) (e.g., \(P(x) = x^2 + 4x + 1\)) * **Degree 3 (Cubic Polynomial):** \(P(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0\) (e.g., \(P(x) = 4x^3 - x + 8\)) * **Degree 4 (Quartic Polynomial):** \(P(x) = a_4 x^4 + \dots + a_0\) * **Degree 5 (Quintic Polynomial):** \(P(x) = a_5 x^5 + \dots + a_0\) #### By Number of Terms: * **Monomial:** A polynomial with one term (e.g., \(7x^3\)) * **Binomial:** A polynomial with two terms (e.g., \(x^2 - 4\)) * **Trinomial:** A polynomial with three terms (e.g., \(2x^3 + 3x - 1\)) ### Operations on Polynomials #### Addition and Subtraction To add or subtract polynomials, combine like terms (terms with the same variable and exponent). **Example:** Let \(P(x) = 3x^3 - 2x + 5\) and \(Q(x) = x^3 + 4x^2 - x - 2\). \(P(x) + Q(x) = (3x^3 - 2x + 5) + (x^3 + 4x^2 - x - 2)\) \( = (3+1)x^3 + 4x^2 + (-2-1)x + (5-2)\) \( = 4x^3 + 4x^2 - 3x + 3\) #### Multiplication To multiply polynomials, use the distributive property, multiplying each term in the first polynomial by each term in the second polynomial, then combine like terms. **Example:** Let \(P(x) = x+2\) and \(Q(x) = x^2 - 3x + 1\). \(P(x) \cdot Q(x) = (x+2)(x^2 - 3x + 1)\) \( = x(x^2 - 3x + 1) + 2(x^2 - 3x + 1)\) \( = x^3 - 3x^2 + x + 2x^2 - 6x + 2\) \( = x^3 + (-3+2)x^2 + (1-6)x + 2\) \( = x^3 - x^2 - 5x + 2\) #### Division (Polynomial Long Division) Polynomial long division is similar to numerical long division. It allows us to divide a polynomial \(P(x)\) by another polynomial \(D(x)\) (where the degree of \(D(x)\) is less than or equal to the degree of \(P(x)\)), to get a quotient \(Q(x)\) and a remainder \(R(x)\). \[ \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} \] Which can also be written as: \[ P(x) = Q(x)D(x) + R(x) \] Where the degree of \(R(x)\) is less than the degree of \(D(x)\). ### Remainder Theorem If a polynomial \(P(x)\) is divided by \((x-a)\), the remainder is \(P(a)\). **Example:** Find the remainder when \(P(x) = x^3 - 2x^2 + 5x - 1\) is divided by \((x-2)\). According to the Remainder Theorem, the remainder is \(P(2)\). \(P(2) = (2)^3 - 2(2)^2 + 5(2) - 1\) \( = 8 - 2(4) + 10 - 1\) \( = 8 - 8 + 10 - 1\) \( = 9\) The remainder is 9. ### Factor Theorem The Factor Theorem is a special case of the Remainder Theorem: * If \((x-a)\) is a factor of \(P(x)\), then \(P(a) = 0\). * Conversely, if \(P(a) = 0\), then \((x-a)\) is a factor of \(P(x)\). **Definition:** A **root** (or **zero**) of a polynomial \(P(x)\) is a value \(a\) such that \(P(a) = 0\). **Example:** Show that \((x+1)\) is a factor of \(P(x) = x^3 + 2x^2 - 5x - 6\). We need to check if \(P(-1) = 0\) (since \(x+1 = x - (-1)\)). \(P(-1) = (-1)^3 + 2(-1)^2 - 5(-1) - 6\) \( = -1 + 2(1) + 5 - 6\) \( = -1 + 2 + 5 - 6\) \( = 0\) Since \(P(-1) = 0\), \((x+1)\) is a factor of \(P(x)\). ### Relationship between Roots and Coefficients (Vieta's Formulas) For a polynomial equation, there are specific relationships between its roots and coefficients. These are known as Vieta's formulas. #### For a Quadratic Equation For a quadratic equation \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\): * Sum of roots: \(\alpha + \beta = -\frac{b}{a}\) * Product of roots: \(\alpha\beta = \frac{c}{a}\) #### For a Cubic Equation For a cubic equation \(ax^3 + bx^2 + cx + d = 0\) with roots \(\alpha\), \(\beta\), and \(\gamma\): * Sum of roots: \(\alpha + \beta + \gamma = -\frac{b}{a}\) * Sum of products of roots taken two at a time: \(\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\) * Product of roots: \(\alpha\beta\gamma = -\frac{d}{a}\) ## Vocabulary * **Polynomial:** An algebraic expression of one or more terms, each term consisting of a constant multiplied by one or more variables raised to non-negative integer powers. * **Term:** A single number, variable, or product of numbers and variables. * **Coefficient:** The numerical factor of a term. * **Variable:** A symbol (usually a letter) that represents a quantity that can change. * **Degree (of a polynomial):** The highest exponent of the variable in a polynomial. * **Constant Term:** A term in a polynomial that does not contain a variable (its degree is 0). * **Root (or Zero):** A value of the variable for which the polynomial evaluates to zero. * **Factor:** An expression that divides another expression evenly (with no remainder). * **Monomial:** A polynomial with exactly one term. * **Binomial:** A polynomial with exactly two terms. * **Trinomial:** A polynomial with exactly three terms. * **Long Division (Polynomial):** A method for dividing polynomials. * **Vieta's Formulas:** Relationships between the roots and coefficients of a polynomial equation. ## Examples ### Example 1: Identifying Polynomial Properties Consider the polynomial \(P(x) = -5x^6 + 2x^3 - x + 10\). * Degree: 6 * Leading coefficient: -5 * Constant term: 10 * Type by degree: Hexic (or 6th degree) polynomial * Number of terms: 4 ### Example 2: Polynomial Long Division Divide \(P(x) = x^3 - 6x^2 + 11x - 6\) by \(D(x) = x - 1\). ``` x^2 - 5x + 6 ________________ x - 1 | x^3 - 6x^2 + 11x - 6 -(x^3 - x^2) _____________ -5x^2 + 11x -(-5x^2 + 5x) _____________ 6x - 6 -(6x - 6) _________ 0 ``` Thus, \( (x^3 - 6x^2 + 11x - 6) \div (x-1) = x^2 - 5x + 6 \) with a remainder of 0. This also shows that \((x-1)\) is a factor of \(P(x)\). ### Example 3: Using Vieta's Formulas The quadratic equation \(2x^2 - 5x + 3 = 0\) has roots \(\alpha\) and \(\beta\). * Sum of roots: \(\alpha + \beta = -\frac{(-5)}{2} = \frac{5}{2}\) * Product of roots: \(\alpha\beta = \frac{3}{2}\) Consider a cubic equation \(x^3 - 4x^2 + x + 6 = 0\) with roots \(\alpha, \beta, \gamma\). * Sum of roots: \(\alpha + \beta + \gamma = -\frac{(-4)}{1} = 4\) * Sum of products of roots taken two at a time: \(\alpha\beta + \beta\gamma + \gamma\alpha = \frac{1}{1} = 1\) * Product of roots: \(\alpha\beta\gamma = -\frac{6}{1} = -6\) ## Exercises 1. For the polynomial \(P(x) = 2x^5 - 3x^4 + x^2 - 8\), state its degree, leading coefficient, and constant term. 2. Let \(A(x) = 5x^3 - 2x^2 + 7\) and \(B(x) = 2x^3 + x^2 - 4x + 1\). * Find \(A(x) + B(x)\). * Find \(A(x) - B(x)\). 3. Multiply the polynomials: \((x^2 + 3x - 2)(x-4)\). 4. Using polynomial long division, divide \(x^3 - 2x^2 - 5x + 6\) by \((x-3)\). 5. Using the Remainder Theorem, find the remainder when \(P(x) = 2x^4 - 3x^2 + 5x - 1\) is divided by \((x+2)\). 6. Using the Factor Theorem, determine if \((x-1)\) is a factor of \(P(x) = x^4 + 2x^3 - x^2 - 2x\). 7. Given that \(x^3 - 6x^2 + kx - 6\) has \((x-1)\) as a factor, find the value of \(k\). 8. The roots of the quadratic equation \(3x^2 + 7x - 2 = 0\) are \(\alpha\) and \(\beta\). Find the values of: * \(\alpha + \beta\) * \(\alpha\beta\) 9. A cubic equation is given by \(2x^3 - x^2 - 7x + 6 = 0\). Its roots are \(\alpha, \beta, \gamma\). Find: * \(\alpha + \beta + \gamma\) * \(\alpha\beta + \beta\gamma + \gamma\alpha\) * \(\alpha\beta\gamma\) ## Assignment 1. Given \(P(x) = 3x^4 - 5x^3 + 2x - 1\) and \(Q(x) = x^4 + 2x^3 - 4x^2 + 5\). * Calculate \(P(x) + 2Q(x)\). * Calculate \(P(x) \cdot Q(x)\). 2. Divide \(2x^4 + 3x^3 - 7x^2 - 11x - 3\) by \((2x+1)\) using polynomial long division. 3. When \(P(x) = x^3 + ax^2 + bx - 12\) is divided by \((x-2)\), the remainder is -8. When it is divided by \((x+1)\), the remainder is -27. Find the values of \(a\) and \(b\). 4. Show that \((x-2)\) is a factor of \(f(x) = x^3 - x^2 - 4x + 4\). Hence, find all the roots of \(f(x) = 0\). 5. The roots of the equation \(x^3 - 5x^2 + qx + r = 0\) are \(\alpha, \beta, \gamma\). If \(\alpha\beta = -2\) and \(\alpha + \beta + \gamma = 5\), find the values of \(q\) and \(r\). 6. Form a cubic equation with integer coefficients whose roots are \(1, 2, -3\). ## Summary In this lesson, we established a strong foundation in polynomials. We defined polynomials by their general form, degree, and number of terms, and learned how to perform basic operations such as addition, subtraction, and multiplication. Polynomial long division was introduced as a method for dividing polynomials. Key theorems for understanding polynomial roots and factors were covered: * The **Remainder Theorem** states that the remainder of \(P(x)\) divided by \((x-a)\) is \(P(a)\). * The **Factor Theorem**, a direct consequence, states that \((x-a)\) is a factor of \(P(x)\) if and only if \(P(a) = 0\). Finally, we explored **Vieta's Formulas**, which provide powerful relationships between the roots and coefficients of quadratic and cubic polynomials. These formulas are incredibly useful for solving problems involving roots without explicitly knowing their values. Mastering these concepts is essential for further studies in algebra and calculus, opening doors to understanding more complex mathematical models.