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Chapter 7: Problem 28

Expand each expression. $$ (3 f+10)(9 f-1) $$

Short Answer

Expert verified
27f^2 + 87f - 10

Step by step solution

01

Apply the Distributive Property

Use the distributive property, also known as the FOIL method (First, Outer, Inner, Last) to multiply each term in the first binomial with each term in the second binomial:y = (3f+10)(9f-1)
02

Multiply the First Terms

Multiply the first terms of each binomial:\[3f \times 9f = 27f^2\]
03

Multiply the Outer Terms

Multiply the outer terms of each binomial:\[3f \times (-1) = -3f\]
04

Multiply the Inner Terms

Multiply the inner terms of each binomial:\[10 \times 9f = 90f\]
05

Multiply the Last Terms

Multiply the last terms of each binomial:\[10 \times (-1) = -10\]
06

Combine Like Terms

Combine the results of the multiplications from Steps 2 to 5:\[27f^2 - 3f + 90f - 10\]Combine the like terms \by adding \[ -3f + 90f = 87f \] Thus resulting in: \[27f^2 + 87f - 10\]

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Key Concepts

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

Expanding Binomials
Expanding binomials involves multiplying two binomial expressions together to create a polynomial. In our example, we expand \( (3f+10)(9f-1) \). The goal is to multiply each term in the first binomial by each term in the second binomial.
For a quick refresher, a binomial is an expression with two separate terms, such as \( a + b \). When we multiply two binomials, we'll have four separate products to combine.
This process ensures we consider all possible pairs of terms from each binomial.
FOIL Method
The FOIL method is a specific way of expanding binomials. FOIL stands for First, Outer, Inner, and Last, which describes the pairs of terms to multiply. Here's how it works:
  • First: Multiply the first terms from each binomial: \( 3f \times 9f = 27f^2 \)
  • Outer: Multiply the outer terms: \( 3f \times (-1) = -3f \)
  • Inner: Multiply the inner terms: \( 10 \times 9f = 90f \)
  • Last: Multiply the last terms: \( 10 \times (-1) = -10 \)
By following these steps, we ensure that all term pairs are accounted for and multiplied correctly.
This systematic approach helps prevent errors and ensures clarity in the multiplication process.
Combining Like Terms
After using the FOIL method, we need to combine like terms to simplify the expression. Like terms are terms that have the same variable raised to the same power.
In our example, the multiplication resulted in: \( 27f^2 - 3f + 90f - 10 \).
Here, \( -3f \) and \( 90f \) are like terms because they both contain the variable \( f \) raised to the first power.
To combine them, we add their coefficients: \( -3f + 90f = 87f \).
Our combined and simplified expression is thus: \[27f^2 + 87f - 10 \]
This final step is crucial as it gives us a cleaner, simpler polynomial.

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Most popular questions from this chapter

Life Science The table lists the approximate amount of dry plant material that each type of habitat produces in 1 year. $$ \begin{array}{ll}{} & {\text { Plant Material (grams) }} \\ {\text { Habitat }} & {\text { Produced per Square Meter }} \\ {\text { Coral reef }} & {2,500} \\ {\text { Tropical rainforest }} & {2,200} \\ {\text { Temperate rainforest }} & {1,250} \\ {\text { Savannah }} & {900} \\ {\text { Open sea }} & {125} \\ {\text { Semidesert }} & {90}\end{array} $$ a. One gram is equivalent to 0.035 ounce, and 1 meter is equivalent to 1.09361 yards. Determine how many ounces of plant material per square yard the typical coral reef produces in 1 year. Show how you found your answer. b. Write two different statements comparing the amount of plant material produced in a tropical rain forest to that produced in a temperate rain forest. c. The Baltic Sea covers approximately \(422,000\) square kilometers. Approximately how many kilograms of plant material are produced in the Baltic Sea in a year? Show how you found your answer.

Solve each equation by factoring using integers, if possible. If an equation can't be solved in this way, explain why. $$ g^{2}+64=16 g $$

Solve each equation. $$ 4(2 z+3)^{2}-2=-1 $$

History When the famous German mathematician Gauss was a young boy, he amazed his teacher by rapidly computing the sum of the integers from 1 to 100. He realized that he could compute the sum without adding all the numbers, by grouping the 100 numbers into pairs. To see a shortcut for finding this sum, look at two lists of 1 to 100, one in reverse order. \(\begin{array}{cccccccccccc}{1} & {2} & {3} & {4} & {5} & {6} & {7} & {\dots} & {50} & {\dots} & {94} & {95} & {96} & {97} & {98} & {99} & {100} \\ {100} & {99} & {98} & {97} & {96} & {95} & {94} & {\dots} & {51} & {\dots} & {7} & {6} & {5} & {4} & {3} & {2} & {1}\end{array}\) a. What is the sum of each pair? b. How many pairs are there? c. What is the sum of all these pairs? d. How many times is each of the integers from 1 to 100 counted in this sum? e. Consider your answers to Parts c and d. What is the sum of the integers from 1 to 100? f. Explain how you can use this same reasoning to find the sum of the integers from 1 to n for any value of n. Write a formula for s, the sum of the first n positive integers. g. Chloe added several consecutive numbers, starting at 1, and found a sum of 91. Write an equation you could use to find the numbers she added. Solve your equation by completing the square. Check your answer with the formula.

Solve each equation. Be sure to check your answers. (Hint: Try factoring the numerator first.) $$ \frac{16 x^{2}-81}{4 x+9}=31 $$
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