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Chapter 5: Problem 54

Find each product. Use the FOIL method. $$ (b+8)(6 b-2) $$

Short Answer

Expert verified
\((b + 8)(6b - 2) = 6b^2 + 46b - 16\)

Step by step solution

01

Identify the terms to be multiplied

The terms in the first binomial are \(b\) and 8. The terms in the second binomial are \(6b\) and -2.
02

Multiply the First terms

Multiply the first terms of each binomial: \(b \times 6b = 6b^2\).
03

Multiply the Outer terms

Multiply the outer terms of each binomial: \(b \times -2 = -2b\).
04

Multiply the Inner terms

Multiply the inner terms of each binomial: \(8 \times 6b = 48b\).
05

Multiply the Last terms

Multiply the last terms of each binomial: \(8 \times -2 = -16\).
06

Combine like terms

Add all the terms from the previous steps: \(6b^2 - 2b + 48b - 16\). Combine the like terms \(-2b\) and \(48b\): \(6b^2 + 46b - 16\).

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Key Concepts

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

Binomials
A binomial is a type of polynomial that contains exactly two terms. For instance, in the given exercise, the expressions \(b+8\) and \(6b-2\) are both binomials. Binomials are important for many algebraic operations and are often used in problems involving the FOIL method. To identify binomials, simply look for expressions with two distinct parts, typically separated by a plus or minus sign. Understanding binomials is the first step in mastering polynomial multiplication.
Polynomials
A polynomial is a mathematical expression composed of variables and coefficients, linked by operations of addition, subtraction, and multiplication. Polynomials can have any number of terms, meaning they can range from monomials (one term) to binomials (two terms) and beyond. In this exercise, we deal with the multiplication of two binomials resulting in a polynomial with multiple terms: \(6b^2, -2b, 48b, -16\). Polynomials are fundamental in algebra and can describe a wide range of mathematical problems and models.
Combining Like Terms
In algebra, combining like terms is a crucial step to simplify polynomial expressions. Like terms are terms that have the same variable raised to the same power. For example, \(-2b\) and \(48b\) are like terms because they both contain the variable \(b\). To combine them, simply add or subtract their coefficients: \(-2b + 48b = 46b\). In the given solution, combining like terms transforms the expression \(6b^2 - 2b + 48b - 16\) into a simpler form: \(6b^2 + 46b - 16\). This step makes the polynomial easier to interpret and work with in further mathematical operations.

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Most popular questions from this chapter

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 201 \times 199 $$

If it costs \(\left(4 x^{5}+3 x^{4}+2 x^{3}+9 x^{2}-29 x+2\right)\) dollars to fertilize a garden, and fertilizer costs \((x+2)\) dollars per square yard, write an expression, in square yards, for the area of the garden.

Graph each equation by completing the table of values. \(y=x^{2}-6\) \(\begin{array}{|c|c|}\hline x & {y} \\ \hline-2 & {} \\ \hline-1 & {} \\\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline\end{array}\)

Assume that \(a\) is a number greater than 1 . Arrange the following terms in order from least to greatest: \(-(-a)^{3},-a^{3},(-a)^{4},-a^{4} .\) Explain how you decided on the order.

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(4 a^{2} b^{3}\right)^{-2}\left(2 a b^{-1}\right)^{3}}{\left(a^{3} b\right)^{-4}} $$
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