Problem 50 - Algebra 1: Concepts and Skills

Question

COMMON FACTOR Factor the expression. $$ 2 x^{2}+28 x y+98 y^{2} $$

Step-by-Step Solution

Answer

The factorized form of the expression \(2x^{2}+28xy+98y^{2}\) is \(2(x+7y)^{2}\).

Step 1: Identify the Common Factor

Look at all the terms in the expression \(2x^{2}+28xy+98y^{2}\) to identify a common factor. In this case, we can see that \(2\) is a common factor among all the terms.

Step 2: Factor out the Common Factor

Factor out the common factor, which in this case is 2. So the expression becomes \(2(x^{2}+14xy+49y^{2})\).

Step 3: Recognize and Factor the Quadratic

The expression inside the parenthesis \(x^{2}+14xy+49y^{2}\) is a quadratic trinomial. We should try to express it as the square of a binomial. By trial, error and observation we obtain \(x^{2}+14xy+49y^{2}=(x+7y)^{2}\).

Step 4: Write the Final Factorized Expression

Replace the quadratic trinomial in the expression with its factorized form to get the final answer: \(2(x+7y)^{2}\).

Key Concepts

Common Factor

Finding a common factor is often the first step when factoring expressions. It simplifies the expression by reducing its terms. In the expression \(2x^{2} + 28xy + 98y^{2}\), the common factor is \(2\). Here’s how you spot it:

• Look at all numerical coefficients in the terms: \(2, 28, 98\).
• Find the greatest number that divides all these coefficients. In this case, it is \(2\).

Once identified, the common factor can be factored out. For our example, when you factor \(2\) out of the entire expression, you are left with \(2(x^{2} + 14xy + 49y^{2})\). Factoring out the common factor helps pave the way for further simplification.

Quadratic Trinomial

A quadratic trinomial typically has three terms with degrees that appear in descending order. The expression within the parentheses: \(x^{2}+14xy+49y^{2}\) is a good illustration of a quadratic trinomial. Here's what makes it one:

• The first term, \(x^2\), is quadratic because of the square power on \(x\).
• The middle term, \(14xy\), contains both variables \(x\) and \(y\), contributing a linear mix.
• The last term, \(49y^2\), is a perfect square with too a degree of 2.

Recognizing this pattern is essential because quadratic trinomials often have a recognizable structure that can transform into a more simplified form during the factoring process.

Binomial Square

A binomial square refers to an expression that is the square of a binomial expression. In the context of the expression \(x^{2}+14xy+49y^{2}\), we recognize it can be rewritten as a binomial square, \((x+7y)^{2}\).This form emerges because:

• The term \((x)^{2}\) is the square of \(x\).
• The term \((7y)^{2}\) is the square of \(7y\).
• The middle term, \(14xy\), is twice the product of the first term \(x\) and the last term \(7y\) of the binomial.

Understanding binomial squares is useful when simplifying or factoring expressions as it reveals an underlying symmetry that is more manageable. Once recognized, this can assist in quickly rewriting and solving expressions by reference to this characteristic perfect square form.