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Chapter 5: Problem 9

Which pair of integers would be used to rewrite the middle term of \(12 y^{2}+5 y-2\) when factoring by grouping? A. -8,3 B. 8,-3 C. -6,4 D. 6,-4

Short Answer

Expert verified
The integers are 8 and -3 (Option B).

Step by step solution

01

- Understand the Problem

We need to find two integers that multiply to the product of the leading coefficient (12) and the constant term (-2), and add up to the middle coefficient (5).
02

- Calculate the Product

The product of the leading coefficient and the constant term is: \( 12 \times (-2) = -24 \)
03

- Identify Required Sum

We need to find two integers that add up to the middle coefficient, which is 5.
04

- Find the Integer Pair

Determine which pair of integers from the options multiply to -24 and add to 5. Option B (8, -3) fits because: \( 8 \times (-3) = -24 \) and \( 8 + (-3) = 5 \)
05

- Verify the Answer

Double-check the calculations to confirm the integer pair is correct. Option B (8, -3) is indeed the correct pair.

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Key Concepts

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

Factoring by Grouping
When solving quadratic equations, one effective method is factoring by grouping. This technique is useful when standard methods, like simple factorization, are not straightforward. The aim is to break down the middle term into two numbers that can simplify the equation further. For example, if we have the quadratic equation in the form of \( ax^2 + bx + c \), our goal is to express it as \(ax^2 + px + qx + c \). Then, we group terms and factor each group separately.
Here鈥檚 a step-by-step of how to apply factoring by grouping:
  • First, identify two numbers whose product is \( ac \) (the product of the leading coefficient and the constant term) and whose sum is \( b \) (the middle coefficient).
  • Next, rewrite the middle term using these two numbers.
  • Group the terms in pairs and factor out the common factor from each pair.
  • Finally, factor out the common binomial factor.
This method may seem tricky initially but makes it easier to break down complex polynomials into simpler factors.
Quadratic Equations
Quadratic equations are polynomial equations of the second degree, generally in the form \( ax^2 + bx + c = 0 \). They are characterized by the presence of the square of the variable (\(x\)), making them non-linear. These equations can be solved using several methods: factoring, completing the square, and the quadratic formula. Each method has its own advantages depending on the equation:
  • Factoring: Best for equations where the quadratic can be easily expressed as a product of binomials.

  • Completing the Square: Useful if the quadratic is difficult to factor.

  • The Quadratic Formula: A universal method applicable to all quadratic equations. Given by \[ x = \frac{-b \, \pm \sqrt{b^2-4ac}}{2a} \], it always provides the solution(s) to the equation.
Understanding the properties and methods to solve quadratic equations is fundamental, as they are foundational in algebra and appear frequently in various areas of mathematics.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into products of simpler polynomials. This process is crucial for simplifying expressions and solving polynomial equations. For a quadratic polynomial like \( 12y^2 + 5y - 2 \), the goal is to express it as a product of binomials. Here鈥檚 a quick guide:
  • Begin by identifying the product of the leading coefficient and the constant term.

  • Find pairs of integers that multiply to this product and add up to the middle coefficient.

  • Rewrite the middle term using these integers and group the terms.

  • Factor each group separately and then factor out the common binomial factor.
This step-by-step approach simplifies complex problems and can invariably lead to the simplest form of the polynomial. Factorizing is not just a tool for solving equations but also for understanding the deeper structure and properties of polynomials.

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Most popular questions from this chapter

Factor each polynomial completely. \(m^{2}-p^{2}+2 m+2 p\)

Factor each binomial completely. \(125 m^{3}+8 p^{3}\)

Solve each problem. The product of the first and third of three consecutive integers is 3 more than 3 times the second integer. Find the integers.

Solve each problem. If an object is projected upward from ground level with an initial velocity of \(64 \mathrm{ft}\) per sec, its height \(h\) in feet \(t\) seconds later is given by $$ h=-16 t^{2}+64 t $$ (a) After how many seconds is the height \(48 \mathrm{ft} ?\) (b) The object reaches its maximum height 2 sec after it is projected. What is this maximum height? (c) After how many seconds does the object hit the ground? (Hint: When the object hits the ground, \(h=0 .\) ) (d) Only one of the two solutions from part (c) is appropriate here. Why? (e) After how many seconds is the height \(60 \mathrm{ft}\). (f) What is the physical interpretation of why part (e) has two answers?

Solve each problem. If an object is projected from a height of \(48 \mathrm{ft}\) with an initial velocity of \(32 \mathrm{ft}\) per sec, its height \(h\) in feet after \(t\) seconds is given by $$ h=-16 t^{2}+32 t+48 $$ (a) After how many seconds is the height \(64 \mathrm{ft}\) ? (Hint: Let \(h=64\) and solve. (b) After how many seconds is the height \(60 \mathrm{ft} ?\) (c) After how many seconds does the object hit the ground? (Hint: When the object hits the ground, \(h=0 .\) ) (d) Only one of the two solutions from part (c) is appropriate here. Why?
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