Question

How do you factor $- 7{x^2} + 175$ ?

Answer 1

Hint: First of all, the given expression is a polynomial of degree $2$. Also, the given expression contains two terms. Now, to factorize the expression, we check whether there is any factor common in between them to take it out. This way we can simplify it further.

Formula used:
${a^2} - {b^2} = (a + b)(a - b)$

Complete step-by-step answer:
We need to factorise the given expression. We know that the given expression is a polynomial in variable $x$.
Also, the exponent of the highest degree term in a polynomial is known as its degree. Therefore, we can see that the degree of the given polynomial is $2$.
The polynomial of degree $2$ is called a quadratic polynomial.
Now, let us take the given expression as
$f(x) = - 7{x^2} + 175$ .
We represented the given expression as $f(x)$. Also, the given expression is a monomial, because a polynomial having two terms is a monomial.
Now, to factorize $f(x)$, we check whether there is any common factor in between the two terms, then take that factor out from both the terms. We get,
$= - 7({x^2} - 25)$ , $7$ is the common factor of both the terms, and in order to make the coefficient of ${x^2}$ positive, we take out $- 7$ common from both the terms.
$= - 7({x^2} - {5^2})$ , now we write $\;25$ as ${5^2}$, because then we will be able to apply the identity, ${a^2} - {b^2} = (a + b)(a - b)$
$= - 7(x + 5)(x - 5)$, this way by using the above mentioned identity, we factorised the given expression.

∴ $f(x) = - 7(x + 5)(x - 5)$ , is the factored form of the given polynomial.

Note:
The given expression is a quadratic polynomial because the degree of the given polynomial is $2$ . The name ‘quadratic’ has been derived from ‘quadrate’, which means ‘square’. Generally, any quadratic polynomial in variable $x$ with real coefficients is of the form $f(x) = a{x^2} + bx + c$ , where $a$ , $b$ and $c$ are real numbers and $a \ne 0$ .