Question

How do you write the expression $\left( x+6 \right)\left( x-2 \right)$ as a polynomial in a standard form.

Answer 1

Hint: Here as you can see that we have to write the given expression in the form of a standard polynomial. Standard form of a polynomial is \[a{{x}^{2}}+bx+c=0\] ,converting the given expression into this standard form.
A polynomial function $f(x)$ of degree $'n'$ is of the form.
\[\Rightarrow \]$f(x)={{a}_{n}}{{x}^{n}}+{{a}_{n}}-1{{x}^{n-1}}+...+{{a}_{1}}x+{{a}_{0}}$
Where an is a non-zero constant, and ${{a}_{n-1}},{{a}_{n-2}},....,{{a}_{0}}$ are any constant. To write the expression $\left( x+6 \right)\left( x-2 \right)$ in standard form you can use (First outer inner last)
It is a standard method of multiplying two binomials. Hence the method is referred to as FOIL method. Here FOIL is refer to F-first term, O- outer term, I- inner term, L- last term
Here you have to multiply this different type of term and in this standard form i.e. \[a{{x}^{2}}+bx+c=0\]

Complete step by step solution:
As you know that given expression are,
\[\Rightarrow \]$\left( x+6 \right)\left( x-2 \right)$
To write the expression in standard form you can use FOIL method with referred as (First Outer Inner Last)
\[\Rightarrow \]First: $x\times x={{x}^{2}}$
\[\Rightarrow \]Outer term are: $x\times -2=-2x$
Inner term: $6\times x=6x$
\[\Rightarrow \]Last term: $6\times \left( -2 \right)=-12$
After foiling, completely write the answer in the form of an equation.
\[\Rightarrow \]${{x}^{2}}-2x+6x=12$
Combine like terms, and find an equation will be.
\[\Rightarrow \]${{x}^{2}}+4x-12$

Therefore, the standard form of given expression is ${{x}^{2}}+4x-12$

Additional Information:
FOIL is a standard method of multiplying two binomials here. The word FOIL is an acronym for the four terms of the product.
First (‘first’ terms of each binomial are multiplied together)
Outer: (Outside terms are multiplied that is the first term of the first binomial and the second term of the second)
Inner: (‘Inside’ terms are multiplied- second term of the first binomial and first term of the second)
Last: (‘Last’ term of each binomial are multiplied)
For example: $\left( 3x+5 \right)\left( 2z+7 \right)$
\[\Rightarrow \]First: $3z\times 2z=6{{z}^{2}}$
\[\Rightarrow \]Outside: $3z\times 7=21z$
\[\Rightarrow \]Inside: $5\times 2z=10z$
\[\Rightarrow \]Last: $5\times 7=35$
Standard form simply refers to the format of a mathematical expression where the terms are arranged by decreasing wider degrees. Where the degree is determined by the exponent value of the variable of each term.
For quadratic equation the standard form is $a{{x}^{2}}+bx+c$

Note: Apply the FOIL (first outer inner last) for converting into standard form here FOIL refers to first, outer, inner and last term. Here remember you have to do the multiplication of this following terms and convert it into standard form of polynomial i.e. $a{{x}^{2}}+bx+c=0$
Combine constant and variable terms for calculating the equation.