/*! 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 17 Use the method of your choice to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 17

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$3 x^{2}-17 x+10$$

Short Answer

Expert verified
The factors of the trinomial \(3x^2 - 17x + 10\) are \((3x - 2)\) and \((x - 5)\).

Step by step solution

01

- Identify the coefficient and constant values

The given trinomial is \(3x^2 - 17x + 10\). Here, the coefficient of \(x^2\) is 3, the coefficient of \(x\) is -17, and the constant term is 10.
02

- Find the pair of numbers

We need two numbers that multiply to 30 (product of 3 and 10) and add up to -17. These numbers are -15 and -2.
03

- Factorize the trinomial

Rewrite the middle term as the sum of terms with our pair numbers as coefficients: \(3x^2 - 15x - 2x + 10\). Group the trinomial into two groups to factor by grouping: \((3x^2 - 15x) - (2x - 10)\). Factorizing gives us \(3x(x - 5) - 2(x - 5)\). Now, \(x - 5\) is a common factor. Extract out to get the factors of the trinomial: \((3x - 2)(x - 5)\).
04

- Verify the factorization using the FOIL method

Foiling the result \((3x - 2)(x - 5)\) will verify our factorization. FOIL stands for 'First, Outer, Inner, Last'. Multiply the first terms in each binomial, then the outer, inner and finally the last terms to get: \(3x^2 - 15x - 2x + 10\), which simplifies to \(3x^2 - 17x + 10\), which is the original trinomial. Hence, the factorization is verified.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Coefficient Identification
When factoring trinomials, the first step is to identify the coefficients and constant terms. For the trinomial given, \(3x^2 - 17x + 10\), let's pinpoint these values:
  • The coefficient of \(x^2\) is 3, which tells us how much of \(x^2\) is present.
  • The coefficient of \(x\) is -17, indicating the quantity and direction (negative) of \(x\).
  • The constant term is 10, a standalone number not linked to any variable.
Identifying these values correctly is paramount. It forms the base for the subsequent steps and ensures the numbers used in calculations are correct.
Product-Sum Method
The product-sum method helps us find two numbers that multiply to a specific product and add to a specific sum. In our trinomial, \(3x^2 - 17x + 10\), the product is derived from multiplying the coefficient of \(x^2\) (3) with the constant (10), giving us 30.

Now, we seek two numbers that multiply to 30 and sum to -17. These numbers are -15 and -2. Finding them correctly involves a bit of trial and error, but knowing multiplication tables or practicing with smaller numbers can help.

Once these numbers are found, they become the new coefficients for splitting the middle term.
Factoring by Grouping
Factoring by grouping is an efficient technique for breaking down trinomials. With our numbers, -15 and -2, found using the product-sum method, we rewrite the middle term as two terms:
  • \(3x^2 - 15x - 2x + 10\)
Now, we have four terms, making it easier to group and factor. Group them as follows:
  • \((3x^2 - 15x) - (2x - 10)\)
Next, factor each group:
  • For \(3x^2 - 15x\), factor out a common term \(3x\), giving \(3x(x - 5)\).
  • For \(-2x + 10\), factor out \(-2\), resulting in \(-2(x - 5)\).
Finally, notice \((x - 5)\) is common in both groups, allowing us to factor further into:
  • \((3x - 2)(x - 5)\)
This step separates complex trinomials into more manageable binomials.
FOIL Method
The FOIL method is a systematic approach to double-check our factorization work. FOIL stands for First, Outer, Inner, and Last, representing the order of terms to multiply in each binomial.

For the factors \((3x - 2)(x - 5)\), apply the FOIL method:
  • First: \(3x \times x = 3x^2\)
  • Outer: \(3x \times -5 = -15x\)
  • Inner: \(-2 \times x = -2x\)
  • Last: \(-2 \times -5 = 10\)
Combine these results: \(3x^2 - 15x - 2x + 10\). Simplifying, we retrieve the original trinomial \(3x^2 - 17x + 10\).

This verification ensures that the factorization process is correct and instills confidence in the results obtained from earlier steps.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

As part of a landscaping project, you put in a flower bed measuring 10 feet by 12 feet. You plan to surround the bed with a uniform border of low-growing plants. a. Write a polynomial that describes the area of the uniform border that surrounds your flower bed. (Hint: The area of the border is the area of the large rectangle shown in the figure minus the area of the flower bed.) b. The low-growing plants surrounding the flower bed require 1 square foot each when mature. If you have 168 of these plants, how wide a strip around the flower bed should you prepare for the border?

Use the \(x\)-intercepts for the graph in \(a[-10,10,1]\) by \([-13,10,1]\) viewing rectangle to solve the quadratic equation. Check by substitution. Use the graph of \(y=x^{2}+3 x-4\) to solve \(x^{2}+3 x-4=0\).

If \((x+2)(x-4)=0\) indicates that \(x+2=0\) or \(x-4=0,\) explain why \((x+2)(x-4)=6\) does not mean \(x+2=6\) or \(x-4=6 .\) Could we solve the equation using \(x+2=3\) and \(x-4=2\) because \(3 \cdot 2=6 ?\)

In Exercises \(129-132,\) determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When a factorization requires two factoring techniques, I'm less likely to make errors if I show one technique at a time rather than combining the two factorizations into one step.

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &3 x^{3}-12 x^{2}-15 x=3 x(x+5)(x-1) ;[-5,7,1] \text { by }\\\ &[-80,80,10] \end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.