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Chapter 5: Problem 39

Factor each trinomial. \(36 m^{2}-60 m+25\)

Short Answer

Expert verified
\( (6m - 5)^2 \)

Step by step solution

01

Identify the Form

Observe that the given trinomial is in the form of a quadratic equation: \( ax^2 + bx + c \). In this case, \(a=36\), \(b=-60\), and \(c=25\).
02

Check for Perfect Square Trinomial

Determine if the trinomial is a perfect square trinomial. A perfect square trinomial takes the form \( (mx)^2 + 2mqx + q^2 \) which factors to \((mx+q)^2\).
03

Identify Components of Perfect Square

Compare \( 36 m^2 - 60 m + 25 \) to the perfect square form. As \( a = 36 \) and \( c = 25 \), check if \( b = -2 \sqrt{36} \sqrt{25} = -2 \cdot 6 \cdot 5 = -60 \). Since it matches, the trinomial is a perfect square.
04

Factor the Trinomial

Since the trinomial is a perfect square, factorize it as: \( (6m - 5)^2 \).
05

Verify the Factorization

Finally, expand \((6m - 5)^2 \) to verify: \( (6m - 5)(6m - 5) = 36m^2 - 30m - 30m + 25 = 36m^2 - 60m + 25 \). The factorization is correct.

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Key Concepts

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

quadratic equations
Quadratic equations are a fundamental concept in algebra, representing polynomial equations of degree two. The general form of a quadratic equation is written as \(ax^2 + bx + c = 0\), where:
  • a: the coefficient of the squared term, making it a quadratic term.
  • b: the coefficient of the linear term.
  • c: the constant term.
Through solving quadratic equations, one seeks to find the values of x that satisfy the equation. This can be done by several methods such as factoring, completing the square, or using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Recognizing the structure of quadratic equations is key to deciding the best method to solve them. Notice the form in our example: \(36m^2 - 60m + 25\).Here, \(a = 36, b = -60, \)and\(c = 25\).
perfect square trinomials
A perfect square trinomial is a special type of quadratic equation that can be expressed as the square of a binomial. These trinomials take the form:\((mx + q)^2 = m^2x^2 + 2mqx + q^2\)To identify if a trinomial is a perfect square trinomial, compare its structure to the above form. For the trinomial \(36m^2 - 60m + 25\), we regard:
  • \(a\) as \(36m^2\)
  • \(c\) as \(25\)
  • \(b\) as \(2mq = -60m\)
Breaking \(-60\) into \(2 \cdot 6 \cdot -5\), we see that \(6m\) and \(-5\) are our binomial components, and thus:\((6m - 5)^2 = 36m^2 - 60m + 25\). So it confirms the perfect square property.
factorization
Factorization involves breaking down an expression into products of simpler expressions. In dealing with quadratic equations, perfect square trinomials, like our example \(36m^2 - 60m + 25\), are ideally suited to be factored into binomials. To factor such a trinomial:
  • Identify that the trinomial is a perfect square.
  • Find the square roots of the first and last terms, \(6m\) and \(5\), respectively.
  • Determine the sign of the middle term (\(60m\), hence, negative).
  • Write the binomial as the product squared: \((6m - 5)^2\).
Expand the binomial to verify: \((6m - 5)(6m - 5) = 36m^2 - 60m + 25\).Correct factorization makes solving and simplifying polynomial expressions much easier. Practice steps to become quick at spotting perfect square trinomials and factorization.

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