Question

How do you simplify \[-3(5{{x}^{2}}+2x+9)+x(2x-3)\]?

Answer 1

Hint: Let a monomial be in the form \[a{{x}^{p}}\], where a is a constant coefficient and p is a constant power. In case of multiplying two monomials together: \[\Rightarrow A{{x}^{p}}={{a}_{1}}{{x}^{{{p}_{1}}}}\times {{a}_{2}}{{x}^{{{p}_{2}}}}\]. The coefficients will multiply and powers will sum then, \[A={{a}_{1}}{{a}_{2}}\] and \[p={{p}_{1}}+{{p}_{2}}\]. Hence, \[A{{x}^{p}}={{a}_{1}}{{a}_{2}}{{x}^{{{p}_{1}}}}{{x}^{{{p}_{2}}}}={{a}_{1}}{{a}_{2}}{{x}^{({{p}_{1}}+{{p}_{2}})}}\]. Similarly, in case of multiplying monomials by polynomials also we do the same thing for each term.

Complete step by step answer:
As per the question, we need to simplify \[-3(5{{x}^{2}}+2x+9)+x(2x-3)\]. We need to distribute the monomial outside the parentheses of given polynomials to each term of the polynomial. By splitting up the equation, we can expand \[-3(5{{x}^{2}}+2x+9)+x(2x-3)\] into the following equation:
\[\Rightarrow -3(5{{x}^{2}})\text{ }-3\left( 2x \right)\text{ }\text{ }-3\left( 9 \right)\text{ }+\text{ }x\left( 2x \right)\text{ }+\text{ }x\left( -3 \right)\]
We know that, \[-3(5{{x}^{2}})\] is equal to \[-15{{x}^{2}}\]; \[-3(2x)\] is equal to \[-6x\]; \[x(2x)\] equals to \[2{{x}^{2}}\] and \[x(-3)\] equals to \[-3x\]. By substituting all these terms into the previous equation, we get
\[\Rightarrow -15{{x}^{2}}-6x-27+2{{x}^{2}}-3x\]
Now we need to rearrange the terms in the above equation for simplification. So, we get
\[\Rightarrow -15{{x}^{2}}+2{{x}^{2}}-6x-3x-27\]
Addition of \[-6x\] and \[-3x\] gives \[-9x\], then on substitution we get
\[\Rightarrow -15{{x}^{2}}+2{{x}^{2}}-9x-27\]
And now, we have to take \[-{{x}^{2}}\] term common in the previous equation to get the simplified form. That is, we get
\[\Rightarrow -(15-2){{x}^{2}}-9x-27\]
On subtracting 2 from 15, we get 13. So, by substituting this value in the previous equation, then we get
\[\Rightarrow -13{{x}^{2}}-9x-27\]

Therefore, \[-13{{x}^{2}}-9x-27\] is the required simplified form of the given equation \[-3(5{{x}^{2}}+2x+9)+x(2x-3)\].

Note: While solving such types of questions, we need to take care while calculating the product of two monomials. While calculating, we need to concentrate on the calculations of coefficient product and also about the summation of powers to get the resulting monomial.