What is a polynomial?

A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents. For example, 3x² + 2x - 5 is a polynomial.

What is the degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial 4x³ + 2x² - x + 7, the degree is 3.

How do you add polynomials?

To add polynomials, combine like terms. For example, to add (2x² + 3x + 1) and (x² + 4x + 5), combine the coefficients of like terms: (2x² + x²) + (3x + 4x) + (1 + 5) = 3x² + 7x + 6.

What is the difference between a monomial and a binomial?

A monomial is a polynomial with only one term, such as 5x or 3xy². A binomial has exactly two terms, such as x + 2 or 3a - 4b.

What is the formula for the difference of squares?

The difference of squares formula states that a² - b² = (a - b)(a + b). For example, 9 - 16 can be factored as (3 - 4)(3 + 4) = -1 * 7.

Solve for x: x² - 5x + 6 = 0.

Hint: Factor the quadratic.

To solve for x, factor the equation: (x - 2)(x - 3) = 0. Thus, x = 2 or x = 3.

What is the result of multiplying (x + 2)(x - 3)?

Using the distributive property (FOIL), (x + 2)(x - 3) = x² - 3x + 2x - 6 = x² - x - 6.

What is the polynomial long division process?

Polynomial long division is similar to numerical long division. Divide the leading term of the dividend by the leading term of the divisor, multiply the entire divisor by this result, subtract, and repeat until all terms are divided.

What is the Remainder Theorem?

The Remainder Theorem states that for a polynomial f(x), the remainder of the division of f(x) by (x - c) is equal to f(c). For example, for f(x) = 2x² + 3x - 5, the remainder when divided by (x - 1) is f(1) = 2(1)² + 3(1) - 5 = 0.

If f(x) = 4x² - 3x + 1, what is f(2)?

Hint: Substitute x with 2.

To find f(2), substitute x with 2: f(2) = 4(2)² - 3(2) + 1 = 16 - 6 + 1 = 11.