Factors and roots of a Equation

In mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.We also have factoring calculator to find factors.

The aim of factorization is usually to reduce something to "basic building blocks," such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.Similarly we can find roots of simultaneous quadratic equations
The opposite of factorization is expansion. This is the process of multiplying together factors to recreate the original, "expanded" polynomial.Let's see equations reducible to quadratic form in the following example.


Question:-

Find the factors and roots for

8x³-12x²-2x+3 = 0

Answer:-

Let us rewrite the equation as

8x³-2x-12x²+3 = 0

by Taking 2x as common from from first 2 terms we get

2x(4x²-1)-12x²+3 = 0

by taking 3 as common from last 2 terms we get

2x(4x²-1)-3(4x²-1)=0

By taking (4x²-1) as common we get

(2x-3)(4x²-1)=0

4x²-1 can be written as (2x+1)(2x-1) using a²-b² formula

(2x-3)(2x-1)(2x+1)=0

So factors are (2x-3)(2x-1)(2x+1)

To find the roots we solve each term separately for x we get

x=3/2,1/2,-1/2 (Roots)