Problem 44 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ (2 x+7) a+(2 x+7) b $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial $$c^2 + 10c + 25$$ is $$(c + 5)^2$$.

Step 1: Identify the type of quadratic equation

We have a quadratic equation in the form of $$c^2 + 10c + 25$$, where the coefficients are a=1, b=10, and c=25. Since the coefficients are all positive, we can attempt to factor the quadratic equation using Perfect Square Trinomial method.

Step 2: Determine if the equation is a perfect square trinomial

In order to determine if the given equation is a perfect square trinomial, we need to check if the equation is in the form of $$a^2 + 2ab + b^2$$. In our given equation, $$c^2 + 10c + 25$$, we can see that: - $$a^2 = c^2$$, therefore, $$a = c$$ - $$b^2 = 25$$, therefore, $$b = 5$$ (since b is positive) - $$2ab = 2 \times c \times 5 = 10c$$ Since the equation does indeed follow the form of a perfect square trinomial, we can proceed to the next step.

Step 3: Factor the perfect square trinomial

In order to factor a perfect square trinomial, we can use this formula: $$a^2 + 2ab + b^2 = (a + b)^2$$ So our factored equation would look like this: $$(c + 5)^2$$ Thus, the factored form of the given polynomial $$c^2 + 10c + 25$$ is $$(c + 5)^2$$.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra, and they typically take the form \(ax^2 + bx + c = 0\). In this standard form, \(a\), \(b\), and \(c\) represent coefficients, with \(a\) being non-zero. The variable \(x\) stands for the unknown that we aim to solve. Quadratic equations are characteristic because they represent parabolic relationships when graphed.

Understanding the structure of quadratic equations helps in solving them through methods such as factoring, using the quadratic formula, or completing the square. Solving these equations is essential as they appear in various real-world applications, including physics, engineering, and economics. A quadratic equation can have two, one, or no real solutions. This is determined by the discriminant, \(b^2 - 4ac\), where a positive discriminant indicates two real solutions, zero indicates one, and a negative discriminant suggests no real solutions.

When approaching these equations, identifying specific forms, like perfect square trinomials, can simplify the factoring process, as demonstrated in the exercise.

Perfect Square Trinomials

Perfect square trinomials are a special type of quadratic equation that can make factoring much easier. A trinomial is considered a perfect square if it can be expressed as \(a^2 + 2ab + b^2\). This format is significant because it can be rewritten in a more compact form, \((a + b)^2\). Recognizing this pattern allows us to factor it without lengthy calculations.

In the exercise, the polynomial \(c^2 + 10c + 25\) is a perfect square trinomial.

• The first term, \(c^2\), is \(c\) squared, indicating \(a = c\).
• The last term, 25, is \(5^2\), meaning \(b = 5\).
• The middle term, 10c, matches \(2ab\) because \(2 \times c \times 5 = 10c\).

Once these conditions are verified, factoring becomes straightforward. Instead of dealing with complex calculations, you recognize the structure and apply the perfect square trinomial formula to factor it as \((c + 5)^2\). This technique simplifies many seemingly complicated polynomials.

Factoring Techniques

Factoring is a method used to simplify polynomial expressions and solve quadratic equations. It involves breaking down a composite expression into a product of simpler elements, or factors, that when multiplied together give the original expression. There are several factoring techniques that can be employed depending on the form of the polynomial.

For basic quadratic equations, common techniques include:

• Using common factors: Take a common factor out from each term.
• Factoring by grouping: Rearranging and grouping terms that share a common factor.
• Employing the difference of squares: Recognizing a pattern of \(a^2 - b^2 = (a - b)(a + b)\).
• Factoring perfect square trinomials: Recognizing and factoring trinomials that fit the \(a^2 + 2ab + b^2\) pattern into \((a + b)^2\).

Among these, recognizing perfect square trinomials is particularly useful. In the exercise, by spotting \(c^2 + 10c + 25\) as a perfect square trinomial, we quickly factored it using the method \((a + b)^2\), leading to \((c + 5)^2\). This process illustrates the power of identification and selection of the appropriate factoring technique, simplifying what might otherwise be a more daunting task.