This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. P(x) = a_n x^n + a_n-1 x^n-1 + + a_1 x + a_0 Here, x is the variable, a_i are the coefficients, and n is a non-negative integer representing the degree of the polynomial. • Terms: Each part of the polynomial separated by addition or subtraction is a term. For example, in 3x^2 + 2x - 5, the terms are 3x^2, 2x, and -5. • Degree: The highest exponent of the variable in a polynomial is its degree. For 3x^2 + 2x - 5, the degree is 2. • Monomial: A polynomial with one term (e.g., 5x^3). • Binomial: A polynomial with two terms (e.g., x + 2). • Trinomial: A polynomial with three terms (e.g., x^2 - 3x + 1). Operations with Polynomials: 1. Addition/Subtraction: Combine like terms (terms with the same variable and exponent). Example: (2x^2 + 3x - 1) + (x^2 - x + 4) = (2x^2 + x^2) + (3x - x) + (-1 + 4) = 3x^2 + 2x + 3. 2. Multiplication: Use the distributive property. Multiply each term in the first polynomial by each term in the second. Example: (x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6. 3. Division: Can be done using polynomial long division or synthetic division. Factoring Polynomials: This is the reverse of multiplication. It involves finding polynomials that multiply together to give the original polynomial. Common methods include: • Factoring out the greatest common factor (GCF). • Factoring quadratics (e.g., x^2 + bx + c). • Difference of squares (a^2 - b^2 = (a-b)(a+b)). • Sum/Difference of cubes (a^3 ± b^3).