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Chapter 11: Problem 34

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. $$ y^{2}-y+ $$ _____

Short Answer

Expert verified
Add \(\frac{1}{4}\) to get \((y - \frac{1}{2})^2\).

Step by step solution

01

Identify the binomial structure

The given expression is a binomial of the form \( y^{2} - y \). To make it a perfect square trinomial, we need to add a constant term \( c \) to transform it into \( y^{2} - y + c \).
02

Determine the constant term

A perfect square trinomial is of the form \((b)^2 - 2ab + (a)^2\). In this case, \((2a)^2 = 2\times 1\times a = 1\). So, \(a = \frac{1}{2}\). The constant term should be \((\frac{1}{2})^2 = \frac{1}{4}\).
03

Add the constant to the binomial

The correct constant to add to the binomial \( y^{2} - y \) is \( \frac{1}{4} \). Thus, the trinomial becomes \( y^{2} - y + \frac{1}{4} \).
04

Verify and factor the trinomial

Check the obtained trinomial \( y^{2} - y + \frac{1}{4} \). It can be factored as \((y - \frac{1}{2})^2\) because \((y - \frac{1}{2})(y - \frac{1}{2}) = y^{2} - y + \frac{1}{4} \).

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Key Concepts

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

Understanding Binomials
A binomial is a mathematical expression that consists of two distinct terms connected by either a plus (+) or minus (-) operator. For example, in the exercise given, the binomial is \(y^2 - y\). Binomials are often used as the starting point in algebra for creating more complex expressions, such as trinomials.

The structure of a binomial is fundamentally simple, consisting of a variable raised to a power and another variable term. This simplicity can make binomials an excellent foundation for understanding more complex algebraic concepts.
  • The given binomial \(y^2 - y\) includes a squared term \(y^2\) and a linear term \(-y\).
  • The process of manipulating a binomial, like adding a constant to form a trinomial, is a standard part of polynomial operations.
Factoring Explained
Factoring is a valuable mathematical process that simplifies expressions by breaking them down into products of simpler factors. When it comes to the trinomial formed by the completion of a perfect square, factoring is necessary to verify if the transformation was successful.

After converting the binomial to a trinomial by adding the required constant term, the next step is to check if this trinomial can indeed be factored back into a binomial squared. For example, the trinomial \( y^2 - y + \frac{1}{4} \) can be factored as \((y - \frac{1}{2})^2\).
  • This means the expression \( (y - \frac{1}{2})(y - \frac{1}{2}) \) equals back the trinomial \( y^2 - y + \frac{1}{4} \).
  • Factoring helps verify the solution because it shows that the perfect square trinomial is indeed the square of a binomial.
Grasping Trinomials
Trinomials are algebraic expressions composed of three terms. An essential form of trinomials in algebra is the perfect square trinomial. A perfect square trinomial can be expressed as the square of a binomial.

The process involves identifying the binomial, finding the constant that makes it a perfect square trinomial, and then confirming it through factoring. For instance, in this exercise, beginning with the binomial \(y^2 - y\) leads to the trinomial \( y^2 - y + \frac{1}{4} \) by incorporating the constant term.
  • A perfect square trinomial has the form \((a)^2 - 2ab + (b)^2\) which confirms it can be expressed as \((a-b)^2\).
  • The middle term, often the trickiest part, should satisfy the condition \(-2ab\), where \(a\) and \(b\) are terms derived from the original binomial.
Role of the Constant Term
The constant term is crucial in transforming a binomial into a perfect square trinomial. In the given example, the constant term added is \(\frac{1}{4}\). This term is calculated to ensure the complete trinomial matches the structure of a perfect square.

When a binomial needs to be converted into a perfect square trinomial, the constant term is derived from the square of half the coefficient of the linear term in the binomial. For \(-y\), the coefficient is \(-1\), so you divide it by 2 to get \(-\frac{1}{2}\) and square it to achieve \( \frac{1}{4} \).
  • The constant ensures that the trinomial aligns with perfect square trinomial rules.
  • Without the correct constant term, the trinomial cannot be accurately expressed as the square of a binomial.

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Most popular questions from this chapter

Solve by completing the square. See Section 11.1. $$ y^{2}+6 y=-5 $$

Notice that the shape of the temperature graph is similar to the curve drawn. In fact, this graph can be modeled by the quadratic function \(f(x)=3 x^{2}-18 x+56,\) where \(f(x)\) is the temperature in degrees Fahrenheit and \(x\) is the number of days from Sunday. (This graph is shown in blue.) Use this function to answer Exercises 85 and 86. Use the function given and the quadratic formula to find when the temperature was \(\left.35^{\circ} \mathrm{F} \text { . [Hint: Let } f(x)=35 \text { and solve for } x .\right]\) Round your answer to one decimal place and interpret your result. Does your answer agree with the graph?

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ f(x)=-2(x-4)^{2}+5 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ f(x)=-3(x+2)^{2}+2 $$

Without calculating, tell whether each graph has a minimum value or a maximum value. See the Concept Check in the section. F(x)=3-\frac{1}{2} x^{2}
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