Problem 54 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$25 y^{2}+20 y+4$$

Step-by-Step Solution

Answer

The completely factored form of the given polynomial \(25y^{2}+20y+4\) is \((5y + 2)^2\).

Step 1: Identify the form of the polynomial

The given polynomial \(25y^{2}+20y+4\) is quadratic, which is in the form \(ax^2 + bx + c\). In this case, \(a=25\), \(b=20\), and \(c=4\).

Step 2: Factor using perfect square trinomial

The given polynomial is a perfect square trinomial because \(a = 5^2\), \(c = 2^2\), and \(b = 2 * 5 * 2\). A perfect square trinomial is a trinomial expression that can be written as the square of a binomial, and it has the form \((x + y)^2 = x^2 + 2xy + y^2\). Thus, the trinomial can be factorized as \((5y + 2)^2\).

Step 3: Check the factorization

To check the factorization, multiply \((5y + 2)\) by itself and observe whether the product equals the original polynomial. \((5y + 2)^2 = 25y^2 + 2*(5y*2) + 4 = 25y^2 + 20y + 4\). The product equals the original expression, so the factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression. Recognizing it can make polynomial factorization straightforward. The general form is

• \( (x + y)^2 = x^2 + 2xy + y^2 \)

This form results from squaring a binomial. Let's break this down with an example: if you have \( (a + b)^2 \), then the expression expands to \( a^2 + 2ab + b^2 \). Notice:

• The first term \( a^2 \) and the last term \( b^2 \) are perfect squares.
• The middle term \( 2ab \) is twice the product of the terms inside the binomial.

In our case, \( 25y^2 + 20y + 4 \) is a perfect square trinomial:

• The first term \( 25y^2 \) is \( (5y)^2 \),
• the last term \( 4 \) is \( 2^2 \),
• and the middle term \( 20y \) fits the pattern of \( 2 * 5y * 2 \).

Recognizing these can help you quickly factor the trinomial as

• \( (5y + 2)^2 \).

This makes working with these expressions faster and easier.

Factoring Techniques

When dealing with quadratic expressions, various factoring techniques can be employed. Choosing the right one depends on the form of the polynomial.

• If the polynomial is a perfect square trinomial, the quickest path is to identify it and write it as the square of a binomial. This is exactly what we used in the exercise.
• For other quadratics, such as those not immediately recognizable, different methods like factoring by grouping or the quadratic formula might be needed.
• Always check if a common factor can be factored out first and simplify the expression.

For example, if the polynomial needs more advanced techniques, such as completing the square or using the quadratic formula, it's essential to practice these techniques separately. Remember that practice and familiarity with recognizing patterns can speed up the process of factoring quadratics.

Polynomial Factorization

Polynomial factorization involves breaking down a complex expression into simpler, multipliable components. It allows us to transform polynomials into products of polynomials of lower degree. To factor a polynomial completely, follow these steps:

• Understand the form: Determine whether the polynomial is quadratic or of higher order.
• Identify special patterns: Check for perfect square trinomials, difference of squares, or sum and difference of cubes.
• Use appropriate techniques: Depending on the pattern, apply the right factorization technique.
• Verify your work: Always multiply back to check if you get the original polynomial.

By recognizing recurring patterns, like the perfect square trinomial as in the exercise, you can effectively simplify complex polynomials into more manageable expressions, aiding in solving equations and further mathematical operations.