Q7.4: Solving Polynomial Equations in Factored Form

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, $$4t^2 + 12t$$ is a polynomial.

Factoring a polynomial means expressing it as a product of its factors, which are polynomials of lower degrees. For example, $$4t^2 + 12t = 4t(t + 3)$$ .

The GCF of a polynomial is the largest polynomial that divides all terms of the polynomial without leaving a remainder. For example, the GCF of $$6x^2 + 9x$$ is $$3x$$ .

To solve a polynomial equation in factored form, set each factor equal to zero and solve for the variable. For example, for $$(x-3)(x+5)=0$$ , set $$x-3=0$$ and $$x+5=0$$ .

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

The roots of a polynomial equation are the values of the variable that make the equation equal to zero. They can be found by factoring the polynomial and applying the zero product property.

The standard form of a polynomial is written in descending order of the degree of its terms. For example, $$4t^2 + 12t$$ is in standard form.

A quadratic equation is a polynomial equation of degree 2, typically in the form $$ax^2 + bx + c = 0$$ , where $$a$$ , $$b$$ , and $$c$$ are constants.

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It affects the end behavior of the polynomial's graph.

A polynomial is factored completely when it is expressed as a product of irreducible polynomials, meaning it cannot be factored further over the given number system.