Algebraic Expressions and Equations

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. It does not have an equality sign.

The Zero-Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. For example, if $$(a)(b) = 0$$ , then $$a = 0$$ or $$b = 0$$ .

To factor $$5x^3 - 7x^2$$ , you can take out the common factor, which is $$x^2$$ . The factored form is $$x^2(5x - 7)$$ .

To solve $$2x^2 - 8x = 0$$ , factor out the common term: $$2x(x - 4) = 0$$ . Then, set each factor to zero: $$2x = 0$$ (gives $$x = 0$$ ) and $$x - 4 = 0$$ (gives $$x = 4$$ ). Thus, the solutions are $$x = 0$$ or $$x = 4$$ .

The product of $$(x + 3)(x^2 - x + 4)$$ is $$x^3 + 2x^2 + x + 12$$ .

Factoring an expression means rewriting it as a product of its factors. For example, factoring $$x^2 - 9$$ gives $$(x - 3)(x + 3)$$ .

A polynomial is an algebraic expression that consists of variables raised to whole number powers and coefficients. For example, $$3x^2 + 2x - 5$$ is a polynomial.

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

Set each factor to zero: $$x - 11 = 0$$ (gives $$x = 11$$ ) and $$5x + 1 = 0$$ (gives $$x = -\frac{1}{5}$$ ). Thus, the solutions are $$x = 11$$ or $$x = -\frac{1}{5}$$ .

A monomial is an algebraic expression with only one term, while a polynomial can have multiple terms. For example, $$3x$$ is a monomial, and $$3x + 2x^2 - 5$$ is a polynomial.