Question

How do you simplify \[(x-4)({{x}^{2}}+3x+2)\]?

Answer 1

Hint: We are given an expression consisting of a product of a monomial and a quadratic equation which we have to simplify. We will open the brackets and multiply each term of the monomial with the quadratic equation. Upon solving the resultant terms, we will get the simplified form of the given expression.

Complete step by step answer:
According to the given question, we have to simplify the given expression. We will begin by writing the given expression, we have,
\[(x-4)({{x}^{2}}+3x+2)\]----(1)
We will multiply each terms of the monomial with the bracket of the quadratic equation and we will get it as,
So we will multiply \[x\] and \[-4\] with \[({{x}^{2}}+3x+2)\]. We will get
\[\Rightarrow x({{x}^{2}}+3x+2)-4({{x}^{2}}+3x+2)\]
So, \[x\] will multiply with \[{{x}^{2}}\] to give \[{{x}^{3}}\], \[x\] will multiply with \[3x\] to give \[3{{x}^{2}}\] and with \[2\] to give \[2x\]. Similarly, \[-4\] will multiply with \[{{x}^{2}}\] to give \[-4{{x}^{2}}\], \[-4\] will multiply with \[3x\] to give \[-12x\] and \[-4\] will multiply with \[2\] to give \[-8\].
Multiplying the corresponding terms, we will get,
\[\Rightarrow {{x}^{3}}+3{{x}^{2}}+2x-4{{x}^{2}}-12x-8\]
Now, we will arrange the above expression in terms of the decreasing powers of the terms.
We get,
\[\Rightarrow {{x}^{3}}+3{{x}^{2}}-4{{x}^{2}}+2x-12x-8\]
We can see that for the coefficients of \[{{x}^{2}}\], we have the terms as \[3{{x}^{2}}\] and \[-4{{x}^{2}}\], adding them we get it as \[-{{x}^{2}}\].
Similarly, for the coefficients of \[x\], we have the terms as \[2x\] and \[-12x\], adding the terms we get \[-10x\].
Solving further we get the new simplified expression as,
\[\Rightarrow {{x}^{3}}-{{x}^{2}}-10x-8\]

Therefore, the simplified expression is \[{{x}^{3}}-{{x}^{2}}-10x-8\].

Note: In the above solution, we multiplied the monomial terms with the quadratic equation but we can also do it the other way round, that is, we can multiply the terms of the quadratic equation with the monomial equation and we will still get the same simplified answer. While multiplying the numbers, write the resulting values correctly.