/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 \((2 a+5)\left(3 a^{2}+7 a-9\rig... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 37

\((2 a+5)\left(3 a^{2}+7 a-9\right)\)

Short Answer

Expert verified
6a^3 + 29a^2 + 17a - 45

Step by step solution

01

- Distribute the First Term

Multiply the first term in the first binomial, which is 2a, by each term in the second trinomial. This gives you: \[2a \times 3a^2 + 2a \times 7a + 2a \times (-9)\] This simplifies to: \[6a^3 + 14a^2 - 18a\]
02

- Distribute the Second Term

Now, multiply the second term in the first binomial, which is 5, by each term in the second trinomial. This gives you: \[5 \times 3a^2 + 5 \times 7a + 5 \times (-9)\] This simplifies to: \[15a^2 + 35a - 45\]
03

- Combine Like Terms

Which ultimately gives: \[6a^3 + 29a^2 + 17a - 45\]

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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 crucial algebraic concept. It's all about distributing, or spreading, multiplication over addition or subtraction inside parentheses. When you come across an expression like \(a(b + c)\), the distributive property tells us to multiply \(a\) by both \(b\) and \(c\). This looks like: \[a(b + c) = ab + ac\]. In our problem, \((2a + 5)(3a^2 + 7a - 9)\), we distribute \(2a\) and \(5\) over \((3a^2 + 7a - 9)\). This step-by-step approach helps keep things organized and clear.
Binomial
A binomial is a polynomial with exactly two terms, linked by a plus or minus sign. Each term can be a number, a variable, or a combination of both. For instance, \(3x + 4\) and \(-a^2 + 2b\) are binomials. In our exercise, \(2a + 5\) is the binomial. Understanding binomials is key because they often appear in various algebraic operations, including multiplication.
Trinomial
A trinomial is a type of polynomial with three terms. Examples include \(x^2 + 5x + 6\) or \(6y^2 - 4y + 10\). In our exercise, \(3a^2 + 7a - 9\) is the trinomial. In polynomial multiplication problems, trinomial terms act like individual pieces that each must be multiplied by every term in the binomial using the distributive property.
Combining Like Terms
Combining like terms is the process of adding or subtracting terms with the same variables raised to the same power. This simplifies the polynomial to its most compact form. For instance, \(3x + 4x\) simplifies to \(7x\). In our final solution, we combine terms with the same power of \(a\) from \((6a^3 + 14a^2 - 18a)\) and \((15a^2 + 35a - 45)\), giving \[6a^3 + 29a^2 + 17a - 45\].

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Most popular questions from this chapter

Problem: Use long division to simplify \(\left(x^{2}-8 x+12\right) \div(x-4)\) Incorrect Answer: $$ \begin{array}{r} x - 4 \longdiv { x ^ { 2 } - 8 x + 1 2 } \\ \frac{-\left(x^{2}-4 x\right)}{-12 x+12} \\ \frac{-(-12 x+48)}{-36} \end{array} $$

The pattern for the difference of squares is given as \((a-b)(a+b)=a^{2}-b^{2}\). Is this equivalent to the pattern \((a+b)(a-b)=a^{2}-b^{2} ?\) Explain.

A drafter is making enlargements of a rectangular drawing that preserve the relative width and length of the drawing. The length of the drawing is five- fourths of the width. a. If \(W=\) width, write a polynomial expression in \(W\) that represents the length, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(W\) that represents the perimeter. c. Write a polynomial expression in \(W\) that represents the area.

\(\left(x^{2}-8 x+10\right) \div(x+2)\)

\(\left(d^{2}-169\right) \div(d+13)\)
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