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Chapter 5: Problem 17

Multiply. See Example 1 $$ -2 d^{4}\left(3 d^{3}-3 d^{2}+2 d\right) $$

Short Answer

Expert verified
The product is \(-6d^7 + 6d^6 - 4d^5\).

Step by step solution

01

Distribute the Outside Term

To multiply \(-2d^4 \left(3d^3 - 3d^2 + 2d\right)\), start by distributing the term \(-2d^4\) to each term inside the parentheses: \( -2d^4 \cdot 3d^3 - 2d^4 \cdot 3d^2 + (-2d^4) \cdot 2d\).
02

Multiply Each Pair of Terms

Perform the multiplication for each pair:1. \(-2d^4 \cdot 3d^3 = -6d^{4+3} = -6d^7\)2. \(-2d^4 \cdot (-3d^2) = 6d^{4+2} = 6d^6\)3. \(-2d^4 \cdot 2d = -4d^{4+1} = -4d^5\).
03

Combine All Terms

Combine all the multiplied terms to express the product: \(-6d^7 + 6d^6 - 4d^5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
Imagine you have a term outside of a parenthesis that needs to be multiplied by each term inside the parenthesis. This is where the distributive property comes in handy. For example, in the expression \(-2d^4(3d^3 - 3d^2 + 2d)\), you take the term outside the parenthesis, \(-2d^4\), and distribute it to each term inside the parenthesis.
This means:
  • Multiply \(-2d^4\) with \(3d^3\)
  • Multiply \(-2d^4\) with \(-3d^2\)
  • Multiply \(-2d^4\) with \(2d\)
Take your time with each multiplication. Do it step by step. Apply the distributive property to when you are dealing with terms in parenthesis. It helps in breaking down complex expressions into simpler parts, making calculations manageable.
Exponent Rules
When you multiply powers with the same base, you add their exponents according to exponent rules. This is a crucial step in polynomial multiplication where each term needs careful handling of both coefficients and exponents.
For instance, when you multiply \(-2d^4\) by \(3d^3\), follow the exponent rule:
  • First, multiply the coefficients: \(-2\) and \(3\) give \(-6\).
  • Then, apply the exponent rule: add \(4 + 3 = 7\) to get \(-6d^7\).
Similarly for the other terms:
  • \(-2d^4 (-3d^2) = 6d^{6}\)
  • \(-2d^4 \cdot 2d = -4d^{5}\)
Make sure to accurately add the exponents to avoid calculation mistakes.
Combining Like Terms
After multiplying each term, it's time to combine like terms, though in this case, each term is unique. This step is crucial in simplifying the expression.
Here's how: If any terms end up having the same variable with the same exponent, you would gather them and combine their coefficients. For the provided problem solution of \(-6d^7 + 6d^6 - 4d^5\), notice that each term has a different exponent, so there are no like terms to combine.Combining like terms involves:
  • Add or subtract the coefficients of terms with the same variable and exponent.
  • Ensure all calculations are correct to maintain accuracy.
Even if the terms appear to be different, observing this step ensures you have simplified and checked the expression fully, resulting in the cleanest possible answer.

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Most popular questions from this chapter

Factor each expression. $$ (c-d) r^{3}-(c-d) s^{3} $$

Calculus. In the advanced mathematics course called Calculus, an important polynomial function is $$f(x)=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}$$ Refer to the polynomial on the right side of the equation. A. How many terms does the polynomial have? B. What is the degree of the polynomial? C. What is the coefficient of the fourth term? D. Find \(f(1)\)

Construct a table of values for each polynomial function using the given values for \(x .\) Then graph the function and find its domain and range. $$ \begin{aligned} &f(x)=-x^{2}+2 x+6\\\ &x=-2,-1,0,1,2,3,4 \end{aligned} $$ $$ \begin{array}{|r|l|} \hline x & f(x) \\ \hline-2 & \\ -1 & \\ 0 & \\ 1 & \\ 2 & \\ 3 & \\ 4 & \\ \hline \end{array} $$

Factor each expression. $$ x^{2}+20 x+100-9 z^{2} $$

Explain what is wrong with the following solution. $$ \text { Solve: } \quad x^{2}-x=6 $$ $$ \begin{array}{l} x(x-1)=6 \\ x=6 \text { or } x-1=6 \end{array} $$ $$ x=-7 $$
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