Question

How do you factor the trinomial ${x^2} - 6x - 7$

Answer 1

Hint: This problem under factors of algebraic expression. Polynomial means a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables rise to a non-negative integral power such as $a + bx + c{x^2}$ polynomial adjective. A trinomial is a polynomial with three terms. The trinomial looks like a quadratic expression so we factorise to find the factors by factorisation method and complete step by step explanation.

Complete step-by-step solution:
Let us consider the trinomial ${x^2} - 6x - 7$
The general form of quadratic expression is $a{x^2} + bx + c$
By comparing the equation, we get $a = 1,b = - 6,c = - 7$
The product of this expression is $a \times c = 1 \times - 7 = - 7$
The sum of this expression is $b = - 6$
We need to find the factors by satisfying the condition that the product of two numbers is $ - 7$ and the sum of two numbers is $ - 6$.
By splitting the middle term, factorize the given expression. After factoring the given quadratic expression can be written as a product of two linear factors.
Therefore the factors are,
The sum of roots is $ - 7 + 1 = - 6$
The product of roots is $ - 7 \times 1 = - 7$
Now factorise the given expression
$ \Rightarrow {x^2} - 7x + x - 7$
$ \Rightarrow x(x - 7) + 1(x - 7)$
$ \Rightarrow (x - 7)(x + 1)$

Therefore the factors are $(x - 7)(x + 1)$

Note: The polynomial can differentiate with degree and term. The polynomial having one term is called monomial. The polynomial having two terms is binomial. The polynomial having three terms is called trinomial. The polynomial with the highest degree one is called linear polynomial. The polynomial with highest degree two is called quadratic polynomial. The polynomial with highest degree three is called cubic polynomial. There are three methods to solve quadratic polynomials: factorisation method, formula method and complete square root method.