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What is Solving Polynomial Equations in Factored Form?

If an equation is in factored form, then it would look something like
(x + 2)(x - 1) = 0. To solve this equation, you would refer to the zero product property, which basically states that if two numbers have a product of zero, then one, or both, of the numbers has to equal zero. Therefore, to solve this equation, you would set each factor equal to zero and solve each individual equation, giving you answers of -2 and 1. Checking these answers in the original equation will show that they both work.

Description by Amanda P. (Secondary Math Education major at New England College)

If you solve polynomial equations in factor form you can break down the equation adn make it easier by seeing all the steps.

Description by Jerry L. (Biology major at University of Memphis)

It is taking a polynomial equation, say:

x^2 - 2x - 8 = 0

and factoring it to solve for the unknown variable (x here). Or, in other words:

x^2 - 2x - 8 = (x - 4)(x + 2) = 0

What you get is:

x - 4 = 0 and x + 2 = 0

or

x = 4, x = -2

Description by Harry J. H. (Computer Engineering major at University of Pittsburgh)

Suppose I asked you to multiply a few or more numbers together without telling you which numbers I was thinking of. I bet it would be nearly impossible to come up with the correct answer. Now, suppose I share with you one of the numbers. Although that would be a step in the right direction, you would be no closer to finding the correct answer. Unless...unless the number I shared with you was the very special number zero. In this case, it wouldn't matter what the other numbers were, you would be correct in telling me that the product of all of those numbers must be zero.

The power of this consistency, that anything multiplied by zero is zero, is what makes solving polynomial equations in factored form so worth studying. When presented with a polynomial in factored form, set equal to zero, the solutions of the variable are the numbers that make each factor equal to zero. See when one factor is equal to zero, just like the numbers in the previous example, we can be sure that the equation is sure to hold try. So, start with the first factor and find the value of the variable that makes the factor equal to zero, adding this number to your solution set. Continuing one factor at a time, solve each factor for zero, until you have exhausted all factors and your solution set is complete.

Thanks for reading!

Description by Richard B. (Mathematics for Education major at York University)

Solving Polynomial Equations is so helpful, if we wanna to get the roots of it. It's like getting their minimal expression. so we can work with it so much easier than with the longer one.

Description by Dreea V. (Actuarial Science major at FES Acatlán, UNAM)

Solving polynomial equations in factored form:
Firstly, the degree of the given equation has to be seen. A degree of an equation is used to solve the equations. The given polynomial should be rearranged so that the order is in descending order as per the degree of each term. Once it is arranged in the descending order then we call it a standard form. For example,
x^3 +2x^2 +3 =0 is in standard form.
The simple way to find the factor is the substitution. The following are the steps to factor the polynomial.
Step 1:
Pick up a random number (say x1) and substituting it in the equation say f(x) will result in f(x1) = 0 . Hence we say that x1 is a solution to the f(x) and (x-x1) is a factor to the given polynomial.
Step 2:
Divide the polynomial with the factor that we found in step 1. Once the remainder is 0 we get the quotient. Now the polynomial can be written as f(x) = (x-x1) * g(x). Where g(x) is the quotient and another function.
Step 3:
Now the step 1 and 2 are applied to g(x) and repeated until we get the polynomial factored completely.
Step 4:
The polynomial will be in the form of f(x) = (x-x1) (x-x2) (x-x3) ....... g(x)

Description by Afroze K. (Computer Science major at CVR College of Engineering)

When polynomial equations are in factored form, they happen to be in the Zero-Product Property form which can be simply solved by by taking each quantity (the contains the expression in the parenthesis) and setting that given quantity equal to zero.

Description by Daniel C. (Civil Engineering major at Ohio University)

Polynomials make up all equations. Using the example equation xy (x times y), we can use the zero-product property to solve for the value of real numbers x or y.
xy = 0 if either x = 0 or y = 0

An equation in factored form shows the total equation in a simpler, product form:
(x^2 - x - 6) = (x+2)(x-3)

So given an equation in factored form such as:
(x+2)(x-3) = 0
the equation can be satisfied by two solutions:
(x+2) = 0
x = -2
or
(x-3) = 0
x = 3

Description by Katie C. (Biology major at Boston College)

Why study Solving Polynomial Equations in Factored Form?

Factoring makes polynomials easier to solve.
What factoring does is breaking the problem to individual parts that can be easier to solve.
It's easier to solve (x-3)(x+3) than x^2 - 9
It also makes the student feel more confident solving polynomials if they are broken down than a big equation dealing with different powers.

Description by Stephany M. (Mathematics major at Lone Star College-North Harris)

Factored form gives a lot of information about a polynomial that the distributed form just can't convey. Most importantly, factored form immediately indicates where a function has its zeros; this is helpful in drawing the graph, as well as in the applications I will detail below.

This basic skill has endless applications. Finding the roots of polynomials will never cease to show up in problems dealing with the nature of the world - in ordinary differential equations, the solution to a characteristic equation (a polynomial) gives all possible solutions to the differential equation. In linear algebra, the ability to solve polynomial equations in key when dealing with eigenvalue problems (since the determinants of variable matrices can get quite messy!). Polynomials also show up in Schroedinger Theory (Quantum Mechanics), reaction rates in chemistry, profit maximization in business economics, and much, much more.

Description by Nickhil N. (Computer Science major at University of Pennsylvania)

This topic is useful when you are finding solutions to quadratic equations. When the polynomial is in factored form, it's easy to find the zeros of the equation, by simply setting both quantities equal to zero and solving for x. This is also helpful when you are graphing the quadratic equation, because it will cross the x axis where the zeros occur

Description by Lindsay S. (Math with teaching degree major at Gonzaga )

If a polynomial is factorized, then it is easy to find the roots of the polynomial.

e.g. if P(x)=(x-a)(x-b)(x+c)

Then the roots of P are simply x=a, x=b and x=-c.

Description by Arash A. (Mathematics, Physics major at The University of Western Australia)

Finding the factored form of polynomial equations can be helpful when you are trying to find the areas where the function is equal to zero. These points are also the same points as where the function intercepts the y-axis (since y=0 at those points)

Description by Kirstie B. (Psychology major at Carleton College)

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