Question

How do you write $y = \left( {x - 4} \right)\left( {x - 2} \right)$ in standard form?

Answer 1

Hint: We will use the method of multiplication of variables to solve this question. So, when variables are the same, multiplying them together compresses them into a single factor (variable). When multiplying variables, we multiply the coefficients and variables as usual. If the bases are the same, we can multiply the bases by merely adding their exponents.

Complete step-by-step solution:
Before proceeding with the question, we should understand the concept of multiplication of variables.
If we are using one variable and one constant, then all we have to do is write the constant and variable together without the multiplication sign. Example: $x \times 2 = 2x$, this occurs because x and 2 represent two different amounts.
If we are multiplying a variable with itself then it is simply that variable squared or cubed or whatever power depending upon how many times you multiplied that variable by itself. Example: $x \times x = {x^2}$.
$ \Rightarrow y = \left( {x - 4} \right)\left( {x - 2} \right)$
As we know that $x \times x = {x^2}$ and any constant multiplied by the variable x is simply constant times the variable x. So, applying these we get,
$ \Rightarrow y = x \times x - 4 \times x - 2 \times x + 4 \times 2$
Now simplifying and rearranging equation (2) we get,
$ \Rightarrow y = {x^2} - 6x + 8$

Hence, the equation in standard form is ${x^2} - 6x + 8$.

Note: The three types of polynomial based on terms are discussed as follows:
Monomial: - If the number of terms in a polynomial is one then they are said to be Monomial. E.g., $4x,2y,{x^4}$ etc.
Binomial: - If the number of terms in a polynomial is two then they are said to be Binomial. E.g., $\left( {6 + 2y} \right),\left( {{x^2} + z} \right)$ etc.
Trinomial: - If the number of terms in a polynomial is three then they are said to be Trinomial. E.g., $\left( {6{x^3}y + {z^2} + 6xy} \right),\left( {8{x^4} + z + 16} \right)$ etc.