Polynomials PDF

Sure! Here’s a simple, study-friendly summary of all the topics related to polynomials: ### 1. **Polynomials** A **polynomial** is an expression made up of variables, exponents, and constants, connected by addition, subtraction, and multiplication. The general form looks like this: \[ a_nx^n + a_...

Sure! Here’s a simple, study-friendly summary of all the topics related to polynomials: ### 1. **Polynomials** A **polynomial** is an expression made up of variables, exponents, and constants, connected by addition, subtraction, and multiplication. The general form looks like this: \[ a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \] - **Degree**: The highest exponent of the variable (e.g., degree of \(3x^4 + 2x^2 - 5\) is 4). ### 2. **Binomials** A **binomial** is a polynomial with two terms. For example: \(x^2 - 4\), or \(2x + 3\). ### 3. **Greatest Common Factor (GCF)** The **GCF** is the largest number or expression that divides all terms of a polynomial. Example: For \(6x^3 + 9x^2\), the GCF is \(3x^2\). ### 4. **Zero-Product Property** If \(ab = 0\), then either \(a = 0\) or \(b = 0\). - Used in factoring: If you have an equation like \((x - 3)(x + 2) = 0\), you can set \(x - 3 = 0\) and \(x + 2 = 0\), solving for \(x = 3\) and \(x = -2\). ### 5. **Trinomials in the Form \(x^2 + bx + c\)** To factor trinomials like \(x^2 + bx + c\), find two numbers that multiply to \(c\) and add to \(b\). Example: For \(x^2 + 7x + 12\), find numbers that multiply to 12 and add to 7: \( (x + 3)(x + 4) \). ### 6. **Trinomials in the Form \(ax^2 + bx + c\)** If the coefficient of \(x^2\) is not 1, multiply \(a \cdot c\), and then factor as usual. Example: For \(2x^2 + 7x + 3\), find factors of \(2 \times 3 = 6\) that add up to 7, which are 6 and 1. Then rewrite and group: \[ 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) \] ### 7. **Difference of Two Squares** The **difference of squares** formula is: \[ a^2 - b^2 = (a - b)(a + b) \] Example: \(x^2 - 16 = (x - 4)(x + 4)\). ### 8. **Perfect Square Trinomials** These are trinomials that come from squaring a binomial: \[ (a + b)^2 = a^2 + 2ab + b^2 \] Example: \(x^2 + 6x + 9 = (x + 3)^2\) since \(9 = 3^2\) and \(6x = 2(3)(x)\). ### 9. **Factoring Polynomials with Four Terms (By Grouping)** Group terms and factor each group separately. Example: For \(x^3 + 3x^2 + 2x + 6\), group as: \[ (x^3 + 3x^2) + (2x + 6) = x^2(x + 3) + 2(x + 3) = (x^2 + 2)(x + 3) \] ### 10. **Square of Sums and Differences** - \((a + b)^2 = a^2 + 2ab + b^2\) - \((a - b)^2 = a^2 - 2ab + b^2\) ### 11. **Product of a Sum and Difference** The product of a sum and a difference follows the pattern: \[ (a + b)(a - b) = a^2 - b^2 \] Example: \((x + 5)(x - 5) = x^2 - 25\). ### 12. **Adding and Subtracting Polynomials** To add or subtract polynomials, combine like terms. Example: - Add \( (3x^2 + 2x - 5) + (x^2 - 4x + 6) \): \[ (3x^2 + x^2) + (2x - 4x) + (-5 + 6) = 4x^2 - 2x + 1 \] - Subtract \( (3x^2 + 2x - 5) - (x^2 - 4x + 6) \): \[ (3x^2 - x^2) + (2x + 4x) + (-5 - 6) = 2x^2 + 6x - 11 \] ### **Key Tips:** - Always check if there is a GCF before factoring. - Recognize patterns like difference of squares or perfect squares for easier factoring. - For trinomials, practice finding pairs of numbers that multiply to \(c\) and add to \(b\). - Group terms wisely when factoring by grouping.