Solving Polynomial Equations by Factoring 5th P

A polynomial equation is an equation that involves a polynomial expression, which is a mathematical expression consisting of variables raised to whole number powers and coefficients. For example, $$x^3 - 3x^2 - 5x + 15 = 0$$ is a polynomial equation.

Factoring a polynomial means expressing it as a product of its factors, which are polynomials of lower degrees. For example, $$x^2 - 5x + 6$$ can be factored as $$(x - 2)(x - 3)$$ .

The first step is to set the polynomial equation equal to zero and then factor the polynomial expression on the left side.

The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This is used to find the solutions of factored polynomial equations.

To factor by grouping, you group terms in pairs, factor out the common factor from each pair, and then factor out the common binomial factor.

A difference of cubes is an expression of the form $$a^3 - b^3$$ , which can be factored as $$(a - b)(a^2 + ab + b^2)$$ .

The factored form is $$(x - 7)(x^2 + 7x + 49)$$ .

The roots of a polynomial equation are the values of the variable that make the equation true (i.e., they satisfy the equation). They represent the x-intercepts of the graph of the polynomial.

First, factor out the common term $$x^2$$ : $$x^2(x^2 + 4x + 3) = 0$$ . Then, factor the quadratic: $$x^2(x + 1)(x + 3) = 0$$ . The solutions are $$x = 0, x = -1, x = -3$$ .

To factor a trinomial of the form $$ax^2 + bx + c$$ , find two numbers that multiply to $$ac$$ and add to $$b$$ . Rewrite the middle term using these numbers, then factor by grouping.