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Chapter 6: Problem 73

These exercises involve grouping symbols, factoring by grouping, and factoring sums and differences of cubes. Multiply or divide as indicated. Write each answer in lowest terms. \(\frac{3 a-3 b-a^{2}+b^{2}}{4 a^{2}-4 a b+b^{2}} \cdot \frac{4 a^{2}-b^{2}}{2 a^{2}-a b-b^{2}}\)

Short Answer

Expert verified
\((3 - a - b)(2a + b)\)

Step by step solution

01

- Factor the Numerators

Factor the numerators of the given fractions. For the first numerator, factor by grouping: \(3a - 3b - a^2 + b^2 = -a^2 + b^2 + 3a - 3b = -(a^2 - b^2) + 3(a - b) = -(a + b)(a - b) + 3(a - b) = (a - b)(-a - b + 3) = (a - b)(3 - a - b).\) For the second numerator, factor as a difference of squares: \(4a^2 - b^2 = (2a - b)(2a + b).\)
02

- Factor the Denominators

Factor the denominators of the given fractions. For the first denominator: \(4a^2 - 4ab + b^2 = (2a - b)^2.\) For the second denominator, factor by grouping:\(2a^2 - ab - b^2 = a(2a - b) - b^2 = a(2a - b) - b(2a - b) = (2a - b)(a - b).\)
03

- Write the Factored Expression

Replace the numerator and denominator of each fraction with their factored forms obtained in the previous steps:\(\frac{(a - b)(3 - a - b)}{(2a - b)^2} \times \frac{(2a - b)(2a + b)}{(2a - b)(a - b)}.\)
04

- Simplify the Expression

Look for common factors that can be canceled out from the numerator and denominator. Cancel out \((a - b)\) and \((2a - b)\) terms:\(\frac{3 - a - b}{2a - b} \times \frac{2a + b}{a - b}.\)Now cancel out \((2a - b)\) from the first fraction with \((2a - b)\) in the denominator of the first fraction leaf:\(\frac{3 - a - b}{1} \times \frac{2a + b}{a - b} = (3 - a - b)(2a + b).\)
05

- Write the Final Answer

Multiply the remaining terms to write the final simplified expression: \((3 - a - b)(2a + b).\) This is the simplified form of the given multiplication of fractions.

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Key Concepts

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

Grouping symbols
Grouping symbols such as parentheses \( ... \), brackets \[ ... \] and braces \{ ... \} are essential in algebra. They tell us which operations to perform first. When solving a problem, always work from the innermost grouping outward. For example, in expressions like \(3(a - b)-a^2+b^2\), we must first simplify inside the parentheses. This helps avoid mistakes and ensures accurate results. Grouping symbols are especially valuable when multiple operations are combined or when dealing with complex fractions. Remember: start with what鈥檚 inside the groups first!
Factoring by grouping
Factoring by grouping is a method used to factor polynomials by grouping terms with common factors. Take our example expression, \(3a - 3b - a^2 + b^2\). By rearranging and grouping terms, we get \( -a^2 + b^2 + 3a - 3b\). Factor each group separately: \(-a^2 + b^2\) becomes \(-(a^2 - b^2)\), and \(3a - 3b\) becomes \(3(a - b)\). Finally, factor out the common term \(a - b\). This gives \((a - b)(3 - a - b)\). Factoring by grouping simplifies complex expressions, making them easier to work with. It鈥檚 like breaking down a big problem into smaller, easier pieces.
Factoring sums and differences of cubes
Factoring sums and differences of cubes involves special formulas. These help simplify expressions like \(a^3 + b^3\) and \(a^3 - b^3\). The formulas are as follows:
\ (a^3 + b^3 = (a + b)(a^2 - ab + b^2) \ and \ a^3 - b^3 = (a - b)(a^2 + ab + b^2). \
Let鈥檚 consider the expression \(4a^2 - b^2\). It might look different, but it鈥檚 close to a difference of squares formula. We rewrite it as \((2a)^2 - b^2\), which is \ (2a - b)(2a + b) \. Recognizing these patterns speeds up our work significantly. By mastering these techniques, solving complicated algebraic problems becomes more manageable.

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Most popular questions from this chapter

In the movie Little Big League, young Billy Heywood inherits the Minnesota Twins baseball team and becomes its manager. Before the biggest game of the year, he can't keep his mind on his job because a homework problem is giving him trouble. If Joe can paint a house in \(3 \mathrm{hr}\) and Sam can paint the same house in \(5 \mathrm{hr},\) how long does it take for them to do it together? With the help of one of his players, Billy solves the problem. Solve the following problem using the method of Example \(3 .\) Then solve it using the formula obtained in Exercise \(47 .\) How do the answers compare? A screen printer can complete a \(t\) -shirt order for a Little League baseball organization in 15 hr using a large machine. The same order would take \(30 \mathrm{hr}\) using a smaller machine. How long would it take to complete the order using both machines together?

These exercises involve factoring sums and differences of cubes. Write each rational expression in lowest terms. $$ \frac{x^{3}-27}{x-3} $$

State what \(x\) represents, write an equation, and answer the question. One-third of a number is 2 more than one-sixth of the same number. What is the number?

In each problem, state what \(x\) represents, write an equation, and answer the question. In a certain fraction, the denominator is 6 more than the numerator. If 3 is added to both the numerator and the denominator, the resulting fraction is equivalent to \(\frac{5}{7} .\) What was the original fraction (not written in lowest terms)?

Simplify each complex fraction. Use either method. $$ \frac{\frac{2}{m^{2}}-\frac{3}{m}}{\frac{2}{5 m^{2}}+\frac{1}{3 m}} $$
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