Question

use the information provided to answer part a and part b. consider the polynomial expression ( x^2 + 3(x^2 - 2x + 7) - x(-3x^2 - 4x) ). when the given expression is simplified completely, what is the coefficient of the ( x^2 ) term? type your answer in the box. when the given expression is simplified completely, what is the coefficient of the ( x ) term? type your answer in the box.

Explanation:

Part A: Coefficient of \( x^2 \) term

Step1: Expand the expression

First, expand the terms using the distributive property (also known as the distributive law, \( a(b + c)=ab + ac \)).
For \( 3(x^2 - 2x + 7) \), we get \( 3\times x^2-3\times2x + 3\times7=3x^2-6x + 21 \).
For \( -x(-3x^2 - 4x) \), we get \( -x\times(-3x^2)-x\times(-4x)=3x^3 + 4x^2 \).
So the original expression \( x^2 + 3(x^2 - 2x + 7)-x(-3x^2 - 4x) \) becomes:
\( x^2+3x^2 - 6x + 21+3x^3 + 4x^2 \)

Step2: Combine like terms for \( x^2 \)

Now, combine the like terms for \( x^2 \). The terms with \( x^2 \) are \( x^2 \), \( 3x^2 \), and \( 4x^2 \).
To combine them, we add their coefficients: \( 1 + 3+4 \) (since the coefficient of \( x^2 \) in \( x^2 \) is 1, in \( 3x^2 \) is 3, and in \( 4x^2 \) is 4).
\( 1+3 + 4=8 \)

Step1: Identify the \( x \) terms

From the expanded expression in Part A: \( x^2+3x^2 - 6x + 21+3x^3 + 4x^2 \), the only term with \( x \) is \( - 6x \).

Step2: Determine the coefficient

The coefficient of a term \( ax \) (where \( a \) is a constant) is \( a \). For the term \( -6x \), the coefficient is \( - 6 \).

Answer:

8

Part B: Coefficient of \( x \) term