Question

How do you find the zeros of \[f(x) = (3x - 5)(2x + 7)\]?

Answer 1

Hint: We know that zeros of a polynomial can be defined as the points where the polynomial becomes zero as a whole.
So, we will equate the value of the given polynomial with zero. Then we can find the value of \[x\].

Complete step-by-step solution:
It is given that; the polynomial is \[f(x) = (3x - 5)(2x + 7)\].
Zeros of a polynomial can be defined as the points where the polynomial becomes zero as a whole. A polynomial having value zero is called zero polynomial.
To find the zeros of the polynomial, we will make \[f(x) = 0\]
It implies,
\[(3x - 5)(2x + 7) = 0\]
We know that, if the product of two or more different polynomials is zero, then the value of the polynomials is zero individually.
So, we have,
\[(3x - 5) = 0\] and \[(2x + 7) = 0\]
This implies,
\[x = \dfrac{5}{3}\] and \[x = \dfrac{{ - 7}}{2}\].

Hence, the zeros of \[f(x) = (3x - 5)(2x + 7)\] are \[x = \dfrac{5}{3}\] and \[x = \dfrac{{ - 7}}{2}\].

Note: Zeros of polynomials are the solutions for any given polynomial for which we need to find the value of the unknown variable. It is also known as the roots of the polynomial. If we know the zeros of the polynomial, we can evaluate the value of roots.
An expression of the form \[{{\text{a}}_{\text{n}}}{{\text{x}}^{{\text{n}}\;}} + {\text{ }}{{\text{a}}_{{\text{n}} - {\text{1}}}}{{\text{x}}^{{\text{n}} - {\text{1}}\;}} + {\text{ }} \ldots \ldots {\text{ }} + {\text{ }}{{\text{a}}_{\text{1}}}{\text{x }} + {\text{ }}{{\text{a}}_0}\], where each variable has a constant accompanying it as its coefficient is called a polynomial of degree ‘n’ in variable \[x\].
Each variable separated with an addition or subtraction symbol in the expression is better known as the term.
The degree of the polynomial is defined as the maximum power of the variable of a polynomial.