Chapter A.2, Problem 13E

Solution

To simplify the expression $2x\left(x^2-x+3\right)$ and write the answer in standard form.

  $2x^3-2x^2+6x$

Given:

The expression, $2x\left(x^2-x+3\right)$

Concept Used:

• Like terms are the terms that contain the same variable with similar exponents on it. Constants are like terms as well.
• In any expression, terms are separated by plus or minus signs.
• Coefficients are the constants in front of the variable terms.
• Distributive property: $a\left(b\pm c\right)=a\cdot b\pm a\cdot c$

Calculation:

In order to simplify the given expression, first use the distributive property to expand the expression and then simplify further by combining the like terms as shown below

  $\begin{array}{c}2x\left(x^2-x+3\right)=2x\cdot x^2-2x\cdot x+2x\cdot 3\\ =2x^3-2x^2+6x\end{array}$

Thus, the expression is $2x^3-2x^2+6x$ .

Now, to write the polynomial in standard form first write the degree of each term and then write the term with higher exponents to lowest exponents and then the constant term as shown below:

The degree of the term $2x^3=3$

The degree of the term $2x^2=2$

The degree of the term $6x=1$

So, the standard form is $2x^3-2x^2+6x$