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Chapter 5: Problem 92

Apply the special factoring rules of this section to factor each binomial or trinomial. $$ y^{2}-1.4 y+0.49 $$

Short Answer

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The factored form is \Hashtags allow grouping threads #hashtags.

Step by step solution

01

Identify the type of trinomial

Recognize that the trinomial is of the form \(ax^2 + bx + c\). Here, the given trinomial is \(y^2 - 1.4y + 0.49\) with \(a = 1\), \(b = -1.4\), and \(c = 0.49\).
02

Recognize the perfect square trinomial form

A trinomial \(a^2 - 2ab + b^2\) can be factored into \((a - b)^2\). Notice that \(y^2 - 1.4y + 0.49\) can be compared with this form.
03

Confirm it is a perfect square trinomial

Check if the middle term, \(-1.4y\), fits \(-2ab\). If \(a = y\) and \(b = 0.7\), then \(-2ab = -2 \times y \times 0.7 = -1.4y\), which matches.
04

Factor the trinomial using the special rule

Since it fits the form \(a^2 - 2ab + b^2\), factor it as \((y - 0.7)^2\).

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Key Concepts

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

Perfect Square Trinomial
A perfect square trinomial arises when you expand a binomial squared. It takes the form: \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\). The key is recognizing the pattern and matching it.
For example, in our original problem: \(y^2 - 1.4y + 0.49\), we observed that it matches the pattern \(a^2 - 2ab + b^2\).
Here \(a = y\) and \(b = 0.7\). Once you compare \( -1.4y = -2ab \), you can confirm the perfect square trinomial by verifying \( b = 0.7 \).
Knowing these patterns helps in quickly factoring such trinomials.
Special Factoring Rules
Special factoring rules come in handy when tackling complex algebraic expressions. There are several critical rules to remember:
  • Difference of squares: \( a^2 - b^2 = (a - b)(a + b) \)
  • Perfect square trinomials: \(a^2 + 2ab + b^2 = (a+b)^2 \) or \(a^2 - 2ab + b^2 = (a-b)^2 \)
  • Sum and difference of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) and \(a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)
By mastering these rules, you can more easily break down and solve different polynomial equations. In our exercise, we specifically used the perfect square trinomial rule to factor \(y^2 - 1.4y + 0.49 \) into \((y - 0.7)^2 \).
Binomial Factorization
Binomial factorization is essential in simplifying polynomials. A binomial is an algebraic expression with two terms, like \(a + b\) or \(a - b\). When factoring trinomials into binomials, the goal is to express the trinomial as the product of two binomial terms.
In our problem: \(y^2 - 1.4y + 0.49\), we identified it as a perfect square trinomial. This allowed us to factor it into a binomial squared: \((y - 0.7)^2\).
Practice identifying patterns and using special factoring rules can make solving these problems faster and easier. Recognize whether the trinomial matches patterns like perfect squares or if it can be factored using other special rules.

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Most popular questions from this chapter

Factor completely. \(2 x^{6}+8 x^{5}-42 x^{4}\)

The polynomial \(12 x^{2}+7 x-12\) does not have 2 as a factor. Explain why the binomial \(2 x-6,\) then, cannot be a factor of the polynomial.

What must be true for \(x^{n}\) to be both a perfect square and a perfect cube?

Factor each trinomial completely. $$ 9 t^{2}+24 t r+16 r^{2} $$

Apply the special factoring rules of this section to factor each binomial or trinomial. $$ p^{2}-\frac{1}{9} $$
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