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Chapter 8: Problem 8

Fill in the blanks. a. \(36 y^{2}-49 m^{4}=(\quad)^{2}-(\quad)^{2}\) b. \(125 h^{3}-27 k^{6}=(\quad)^{3}-(\quad)^{3}\)

Short Answer

Expert verified
a. \((6y)^2 - (7m^2)^2\); b. \((5h)^3 - (3k^2)^3\).

Step by step solution

01

Identify the Expression Type for Part (a)

The expression given is \(36 y^2 - 49 m^4\). We can see that both terms are perfect squares: \(36 y^2 = (6y)^2\) and \(49 m^4 = (7m^2)^2\). Thus, this expression can be expressed as a difference of squares.
02

Express as Difference of Squares for Part (a)

Now that we have identified each term as a perfect square, the expression \(36 y^2 - 49 m^4\) can be rewritten as \((6y)^2 - (7m^2)^2\). Thus \((6y)\) and \((7m^2)\) are the terms that fill the blanks.
03

Identify the Expression Type for Part (b)

The expression given is \(125 h^3 - 27 k^6\). Each term is a perfect cube: \(125 h^3 = (5h)^3\) and \(27 k^6 = (3k^2)^3\). So, this expression can be expressed as a difference of cubes.
04

Express as Difference of Cubes for Part (b)

Both terms are perfect cubes, so the expression \(125 h^3 - 27 k^6\) can be rewritten as \((5h)^3 - (3k^2)^3\). Thus \((5h)\) and \((3k^2)\) are the terms that fill the blanks.

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Key Concepts

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

Difference of Squares
The Difference of Squares is a concept in algebra where an expression takes the form of \[a^2 - b^2\]. This expression can be factored into the product of two binomials: \[(a-b)(a+b)\]. This property is based on the foundational identity that the middle terms cancel out, leaving only the subtraction between the two squares.

A practical example of this would be:
  • Take the expression \[36 y^2 - 49 m^4\].
  • We identify the first term, \[36 y^2\], as \[(6y)^2\], and the second term, \[49 m^4\], as \[(7m^2)^2\].
  • By applying the difference of squares identity, we rewrite it as \[(6y - 7m^2)(6y + 7m^2)\].
This simple yet powerful factorization method is valuable in simplifying expressions and solving equations.
Difference of Cubes
The Difference of Cubes involves expressions of the form \[a^3 - b^3\]. It can be factored using the identity: \[(a-b)(a^2 + ab + b^2)\]. Unlike the difference of squares, the factorization includes three terms in one of the factors.

Here鈥檚 how it is applied:
  • Consider the expression \[125 h^3 - 27 k^6\].
  • This expression can be seen as \[(5h)^3 - (3k^2)^3\].
  • Following the difference of cubes rule, it becomes \[(5h - 3k^2)(25h^2 + 15hk^2 + 9k^4)\].
Using this rule helps in simplifying expressions and tackling polynomial equations, providing deeper insights into algebraic structures.
Perfect Squares
A Perfect Square is an expression that results from squaring a binomial. It generally takes the form of \[(a+b)^2 = a^2 + 2ab + b^2\] or \[(a-b)^2 = a^2 - 2ab + b^2\]. Recognizing perfect squares is essential in both expansion and factorization exercises.

Consider the expression \[36 y^2\]. Since \[36 = 6^2\], we identify this as a perfect square \[(6y)^2\]. Seeing perfect squares allows you to simplify even complex expressions or equations, especially when they appear in quadratic equations.
Perfect Cubes
Perfect Cubes refer to expressions derived from raising a number or a term to the power of three. This typically appears as \[a^3\]. A perfect cube is helpful in recognizing patterns in equations, particularly regarding cube roots and factorization.

For example:
  • In the expression \[125 h^3\], we have \[(5h)^3\], thus understanding it as a perfect cube.
  • Similarly, \[27 k^6\] is \[(3k^2)^3\].
Understanding perfect cubes unlocks the potential for more advanced polynomial manipulation, including solving equations via cube roots.

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Most popular questions from this chapter

Roller Coasters. The polynomial function \(f(x)=0.001 x^{3}-0.12 x^{2}+3.6 x+10\) models the path of a portion of the track of a roller coaster. Use the function equation to find the height of the track for \(x=0,20,40,\) and 60.

Storage Tanks. The volume \(V(r)\) of the gasoline storage tank, in cubic feet, is given by the polynomial function \(V(r)=4.2 r^{3}+37.7 r^{2}\) where \(r\) is the radius in feet of the cylindrical part of the tank. What is the capacity of the tank if its radius is 4 feet?

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope \(5, y\) -intercept \((0,-3)\)

Stopping Distances. The number of feet that a car travels before stopping depends on the driver's reaction time and the braking distance. For one driver, the stopping distance \(d(v),\) in feet, is given by the polynomial function \(d(v)=0.04 v^{2}+0.9 v,\) where \(v\) is the velocity of the car in mph. Find the stopping distance at 60 mph. (IMAGE CANNOT COPY)

Perform the operations and simplify, if possible. See Example 6 $$\frac{a^{4} b}{14} \div \frac{a^{3} b^{2}}{21}$$
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