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Chapter 6: Problem 162

Factor completely using the perfect square trinomials pattern. $$ 49 s^{2}+154 s+121 $$

Short Answer

Expert verified
(7s + 11)^2

Step by step solution

01

Identify the Form

Check if the given trinomial fits the pattern of a perfect square trinomial which is \(a^2 + 2ab + b^2\).
02

Find the Square Roots

Determine the square roots of the first and last terms: \(49s^2 = (7s)^2\) and \(121 = 11^2\).
03

Verify the Middle Term

Check if the middle term can be written as \(2ab\). Here, it should be \(2 \times 7s \times 11 = 154s\). This matches the given middle term.
04

Write as a Square of a Binomial

Since the trinomial matches the pattern, write it as the square of a binomial: \((7s + 11)^2\).

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Key Concepts

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

Perfect Square Trinomials
A perfect square trinomial is a special type of polynomial that can be expressed as the square of a binomial. The general form is:
  • \[a^2 + 2ab + b^2\]
  • \[a^2 - 2ab + b^2\]
This means that both the first and third terms are perfect squares, and the middle term is twice the product of their square roots. Recognizing this pattern is key to factoring such trinomials. For example, let's take the given trinomial:
\[49s^2 + 154s + 121\]
Notice that we can write
\[49s^2 = (7s)^2\]
and
\[121 = 11^2\]
Now, confirm that the middle term is twice the product of these square roots:
\[2 \times 7s \times 11 = 154s\]
So, the trinomial fits the perfect square trinomial form and can be factored as:
\[(7s + 11)^2\]
Binomial Expansion
Understanding the concept of binomial expansion helps in identifying and factoring perfect square trinomials. A binomial expansion refers to the algebraic expansion of powers of a binomial. The expansion of
\[(a + b)^2\]
provides a perfect template:
  • \[(a + b)^2 = a^2 + 2ab + b^2\]
  • \[(a - b)^2 = a^2 - 2ab + b^2\]
By expanding the binomial
\[(7s + 11)^2\]
, we get:
\[ (7s + 11)(7s + 11) = 49s^2 + 77s + 77s + 121 = 49s^2 + 154s + 121\]
This shows that
\[49s^2 + 154s + 121\]
is indeed a perfect square trinomial because it can be written as the expansion of
\[(7s + 11)^2\]
Polynomial Factoring
Polynomial factoring is breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. This method simplifies algebraic expressions and solves polynomial equations more easily. For factoring trinomials, we often look for patterns, like the perfect square trinomials.
For example, in the polynomial
\[49s^2 + 154s + 121\]
, we recognize it as a perfect square trinomial. Here's a step-by-step approach to factor it:
  • Identify the squares:
    \[49s^2 = (7s)^2\] and
    \[121 = 11^2\]
  • Verify the middle term:
    \[2 \times 7s \times 11 = 154s\]
  • Since all conditions are met, write the trinomial as a squared binomial:
    \[(7s + 11)^2\]
This approach ensures that the trinomial is fully factored and simplifies solving further algebraic problems.

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Most popular questions from this chapter

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