Chapter 14.4, Problem 31HP

To determine

Toexplain:the product of $x^2-2x+1$ and a monomial $4x^3-8x^2+4x$ and explain the monomial

Answer to Problem 31HP

  $\left(x^2-2x+1\right)\left(4x^3-8x^2+4x\right)=4x^5-16x^4+24x^3-16x^2+4x$ .

Explanation of Solution

Given:

The given product is: $-2nand3-5n$

Calculation:

As thegiven productof $x^2-2x+1$ and a monomial $4x^3-8x^2+4x$ .

This is expanded as

  $\begin{array}{l}\left(x^2-2x+1\right)\left(4x^3-8x^2+4x\right)\{\text{given equation}}\\ =x^2\left(4x^3-8x^2+4x\right)-2x\left(4x^3-8x^2+4x\right)+1\left(4x^3-8x^2+4x\right)\{\text{distributive property}}\\ =4x^5-8x^4+4x^3-8x^4+16x^3-8x^2+4x^3-8x^2+4x\{\text{simplify}}\\ =4x^5-8x^4-8x^4+4x^3+16x^3+4x^3-8x^2-8x^2+4x\{arrangethelike\text{term}}\\ =4x^5-16x^4+24x^3-16x^2+4x\end{array}$

Thus, $\left(x^2-2x+1\right)\left(4x^3-8x^2+4x\right)=4x^5-16x^4+24x^3-16x^2+4x$

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:

1. A monomial, also called power product, is a product of power. of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. The constant 1 is a monomial, being equal to the empty product and  $x^0$  for any variable  $x$ .If only a single variable x is considered, this means that a monomial is either 1 or a power  $x^0$ of x, with n a positive integer.
2. A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is 1.

Hence, $\left(x^2-2x+1\right)\left(4x^3-8x^2+4x\right)=4x^5-16x^4+24x^3-16x^2+4x$ .