Chapter 3: Problem 90
Multiply. $$(3 m-7)\left(m^{2}-5 m-2\right)$$
Short Answer
Expert verified
Answer: The product is 3m^3 - 22m^2 + 29m + 14.
Step by step solution
01
Distribute the term \(3m\)
Multiply the first term of the binomial, \(3m\), with each term of the trinomial \((m^2 - 5m - 2)\). This will give us:
$$(3m) \cdot (m^2) + (3m) \cdot (-5m) + (3m) \cdot (-2)$$
02
Distribute the term \(-7\)
Multiply the second term of the binomial, \(-7\), with each term of the trinomial \((m^2 - 5m - 2)\). This will give us:
$$(-7) \cdot (m^2) + (-7) \cdot (-5m) + (-7) \cdot (-2)$$
03
Add the results of Step 1 and Step 2
Combine the terms obtained in Steps 1 and 2:
$$(3m)(m^2) + (3m)(-5m) + (3m)(-2) + (-7)(m^2) + (-7)(-5m) + (-7)(-2)$$
04
Simplify each term
Perform the multiplication for each term to simplify the expression:
$$3m^3 - 15m^2 - 6m - 7m^2 + 35m + 14$$
05
Combine like terms
Combine the terms containing the same variables raised to the same powers:
$$3m^3 - 15m^2 - 7m^2 - 6m + 35m + 14$$
$$3m^3 - 22m^2 + 29m + 14$$
The final answer for this multiplication is:
$$(3m - 7)(m^2 - 5m - 2) = 3m^3 - 22m^2 + 29m + 14$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributive Property
The distributive property is a cornerstone of algebra and provides a powerful tool in polynomial multiplication. It allows you to multiply a single term by each term within a parenthesis separately before summing up the results.
For instance, when you have an expression like \( (a + b) \times c \), the distributive property enables you to multiply \( c \) by \( a \) and \( b \) individually, resulting in \( ac + bc \). Similarly, when multiplying polynomials, you distribute each term of one polynomial across each term of another polynomial. In the given exercise, we applied the distributive property first with \( 3m \) and then with \( -7 \) to distribute them across \( m^2 - 5m - 2 \).
When you are working on these kinds of multiplication problems, imagine that you're opening up a folded fan—one fold at a time—combining each separate part to display the full picture.
For instance, when you have an expression like \( (a + b) \times c \), the distributive property enables you to multiply \( c \) by \( a \) and \( b \) individually, resulting in \( ac + bc \). Similarly, when multiplying polynomials, you distribute each term of one polynomial across each term of another polynomial. In the given exercise, we applied the distributive property first with \( 3m \) and then with \( -7 \) to distribute them across \( m^2 - 5m - 2 \).
When you are working on these kinds of multiplication problems, imagine that you're opening up a folded fan—one fold at a time—combining each separate part to display the full picture.
Combining Like Terms
After utilizing the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In essence, they are identical in their variable features, although their coefficients can differ.
When you combine like terms, focus on adding or subtracting the coefficients while keeping the variable part constant. For example, \( 3m^2 \) and \( -7m^2 \) are like terms because they both contain the variable \( m \) raised to the second power. Combining these terms in our exercise, \( -15m^2 \) and \( -7m^2 \) combine to form \( -22m^2 \).
Remember to approach this methodically: organize the terms in descending order according to their degree (highest exponent first) and then combine. This technique makes identifying and combining like terms much simpler.
When you combine like terms, focus on adding or subtracting the coefficients while keeping the variable part constant. For example, \( 3m^2 \) and \( -7m^2 \) are like terms because they both contain the variable \( m \) raised to the second power. Combining these terms in our exercise, \( -15m^2 \) and \( -7m^2 \) combine to form \( -22m^2 \).
Remember to approach this methodically: organize the terms in descending order according to their degree (highest exponent first) and then combine. This technique makes identifying and combining like terms much simpler.
Simplifying Expressions
Finally, simplifying expressions is the act of reducing a complex equation into its simplest form. To simplify an algebraic expression, follow a systematic approach: first, distribute the terms as we did with the distributive property, then combine like terms, and finally, perform any operations that will make the expression more concise.
Throughout the process, make sure every operation adheres to mathematical laws, ensuring the integrity of the expression is maintained. In the given polynomial multiplication exercise, we simplified by multiplying the terms and then combining like terms to arrive at the final simplified result, \( 3m^3 - 22m^2 + 29m + 14 \). Additionally, ensuring that your final expression is ordered according to descending powers of the variable provides clarity and is often the conventional way of presenting polynomial expressions.
Throughout the process, make sure every operation adheres to mathematical laws, ensuring the integrity of the expression is maintained. In the given polynomial multiplication exercise, we simplified by multiplying the terms and then combining like terms to arrive at the final simplified result, \( 3m^3 - 22m^2 + 29m + 14 \). Additionally, ensuring that your final expression is ordered according to descending powers of the variable provides clarity and is often the conventional way of presenting polynomial expressions.
