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Chapter 4: Problem 26

Factor. \(y^{4}+14 y^{2}+49\)

Short Answer

Expert verified
The factored form is \((y^2 + 7)^2\).

Step by step solution

01

Identify the Structure

Notice that the expression \( y^4 + 14y^2 + 49 \) resembles a quadratic trinomial but in terms of \( y^2 \). Specifically, it can be written as \((y^2)^2 + 14(y^2) + 49\).
02

Recognize a Perfect Square

Look for patterns that suggest a perfect square trinomial. We check if \( (y^2 + b)^2 = y^4 + 14y^2 + 49 \). The middle term is \( 2ab \) where \( a = y^2 \). Solve for \( b \) with \( 2(y^2)b = 14(y^2) \), giving \( b = 7 \).
03

Verify the Perfect Square Pattern

Verify that \( (y^2 + 7)^2 = y^4 + 14y^2 + 49 \). Expand \( (y^2 + 7)^2 \) to get \( y^4 + 14y^2 + 49 \), confirming the original polynomial matches this expansion.
04

Write the Factored Form

The expression \( y^4 + 14y^2 + 49 \) factors to \((y^2 + 7)^2\). This shows that it is a perfect square trinomial.

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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 is a special type of polynomial. It takes the form \((a + b)^2 = a^2 + 2ab + b^2\). When we deal with these, we are essentially looking to see if a polynomial is a perfect square.
To recognize a perfect square trinomial, notice if the first and last terms are perfect squares. Also, check if the middle term is twice the product of the square roots of those two squares.
  • The expression given is \(y^4 + 14y^2 + 49\).
  • Identify \(y^4\) as \((y^2)^2\) and \(49\) as \(7^2\).
  • To check, see if \(2\times y^2 \times 7\) leads to \(14y^2\).
Indeed, it matches the above requirement, confirming that this is a perfect square trinomial.
Algebraic Expressions
Algebraic expressions involve numbers, variables, and operators. They form the basis of algebra, allowing us to write mathematical relationships and solve equations.
In the case of the expression \(y^4 + 14y^2 + 49\), we are dealing with a polynomial, which is a sequence of terms combined using addition or subtraction. Here:
  • \(y^4\) is a term with a degree of 4.
  • \(14y^2\) is a term combining a coefficient (14) and a square of a variable \(y\).
  • \(49\) is a constant term.
The goal is to rewrite these expressions in different forms that can simplify calculations or solutions.
Quadratic Trinomials
A quadratic trinomial is a polynomial with three terms, in the form of \( ax^2 + bx + c \). In this context, it actually behaves like a squared quadratic due to the structure of the original problem, \( y^4 + 14y^2 + 49 \) resembling the form \((y^2)^2 + 14(y^2) + 49\).
Here's how you can look at it:
  • The term \(y^4\) can be seen as \((y^2)^2\).
  • \(14y^2\) is akin to a linear term in \(y^2\).
  • \(49\) is the constant, or \(c\) in the quadratic form.
Understanding the structure allows for easier identification and factorization of trinomials.

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