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Magazine Chapter 0: Problem 88
Factor the expression. \(25 a^{2}+60 a b+36 b^{2}\)
Short Answer
Expert verified
The expression factors to \((5a + 6b)^2\).
Step by step solution
01
Identify the Form
The expression is in the form of a quadratic trinomial: \(25a^2 + 60ab + 36b^2\). Notice that it fits the pattern \(Ax^2 + Bxy + Cy^2\).
02
Check if it's a Perfect Square Trinomial
To factor the trinomial, check if it is a perfect square. The expression will be a perfect square if \(A\), \(B\), and \(C\) satisfy the equation: \(( rac{B}{2})^2 = AC\). In this case, \(( rac{60}{2})^2 = 25 \times 36\); both equal 900, confirming it is a perfect square.
03
Write as a Perfect Square
Since the trinomial is a perfect square, express it in the form \((D x + Ey)^2\). We'll find \(D\) and \(E\) by taking the square root of \(A\) and \(C\), respectively: \(D = \sqrt{25} = 5\) and \(E = \sqrt{36} = 6\). Thus, the trinomial can be expressed as: \((5a + 6b)^2\).
04
Verify the Factorization
Expand \((5a + 6b)^2\) to verify: \((5a + 6b)(5a + 6b) = 25a^2 + 30ab + 30ab + 36b^2 = 25a^2 + 60ab + 36b^2\). The expression matches the original, confirming the factorization is correct.
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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 kind of quadratic trinomial. It takes on a specific form that allows it to be factored into a squared binomial. Understanding this concept is key to simplifying and solving equations effectively.
In a perfect square trinomial, we have expressions like \[Ax^2 + Bxy + Cy^2\]which can be rewritten as:\[(Dx + Ey)^2\].
Here's how it works:
In a perfect square trinomial, we have expressions like \[Ax^2 + Bxy + Cy^2\]which can be rewritten as:\[(Dx + Ey)^2\].
Here's how it works:
- Identify if the trinomial can be expressed as a square. This involves checking if \[(\frac{B}{2})^2 = AC\],where \(A\), \(B\), and \(C\) are coefficients of the trinomial.
- If this equation holds true, then the trinomial is a perfect square. This was demonstrated in our original problem, where both sides equaled 900.
- Once identified, finding the square roots of \(A\) and \(C\) will help us find \(D\) and \(E\) needed to write the expression as a perfect square.
Quadratic Trinomial
A quadratic trinomial is a polynomial consisting of three terms with the highest degree being two. It is generally expressed in the form \[Ax^2 + Bx + C\], where \(A\), \(B\), and \(C\) are constants.
Quadratic trinomials are a broad category, encompassing many polynomial expressions you encounter in algebra.
Key points include:
Quadratic trinomials are a broad category, encompassing many polynomial expressions you encounter in algebra.
Key points include:
- The main term, \(Ax^2\), signifies that the polynomial is quadratic.
- The linear term, \(Bx\), introduces a single power of \(x\) apart from the quadratic term.
- The constant term, \(C\), is free of any variable.
Polynomial Factorization
Polynomial factorization is a process used in algebra to reduce polynomials into a product of simpler factors. This makes solving polynomial equations more manageable and is a core component of algebra.
Factorization involves these steps:
Factorization involves these steps:
- Identify the structure: Determine the type of polynomial you are working with. For the quadratic trinomial, it will help to see if it is a perfect square, like in our exercise.
- Apply methods: Use methods such as the greatest common factor (GCF), difference of squares, or for specific forms like perfect squares, recognizing and rewriting them appropriately.
- Verify results: After factorizing, multiply the factors to ensure they reconstruct the original polynomial. This ensures accuracy in your factorization.
