Problem 16 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Check your answer. $$j^{2}+9 j+20$$

Step-by-Step Solution

Answer

The completely factored expression for \(j^2 + 9j + 20\) is \((j + 4)(j + 5)\).

Step 1: Identify the quadratic expression

The given quadratic expression is \(j^2 + 9j + 20\).

Step 2: Factor the quadratic expression

We are looking for two numbers that multiply to 20 and add up to 9. The pair of numbers that fit these criteria are 4 and 5. Now, we can rewrite the quadratic expression as follows: \[j^2 + 9j + 20 = j^2 + 4j + 5j + 20\]

Step 3: Group the terms

Next, group the terms with a common factor: \[(j^2 + 4j) + (5j + 20)\]

Step 4: Factor out the common factors

Factor out the common factor from each group: \[j(j + 4) + 5(j + 4)\]

Step 5: Factor the entire expression

Now, we can see that both terms have a common factor of \((j + 4)\). We factor this out to get the completely factored expression: \[(j + 4)(j + 5)\]

Step 6: Check the answer

To check our answer, we multiply the factors back together: \[(j + 4)(j + 5) = j^2 + 5j + 4j + 20 = j^2 + 9j + 20\] Since our factored expression matches the original expression, our answer is correct. The completely factored expression for \(j^2 + 9j + 20\) is \((j + 4)(j + 5)\).

Key Concepts

Quadratic Expressions

Understanding quadratic expressions is essential in mastering mathematics, particularly when it comes to algebra. A quadratic expression is a polynomial of degree two, which often takes the form \(ax^2 + bx + c\). Here, \(x\) is a variable, and \(a\), \(b\), and \(c\) are constants, with \(a\) not equal to zero. The expression \(j^2 + 9j + 20\) is an example of a quadratic expression, where \(j\) represents the variable.Quadratic expressions are fundamental because they appear in many mathematical problems, including physics and engineering contexts. Recognizing a quadratic expression involves identifying the squared term which dictates the expression's degree.

Factoring by Grouping

Factoring by grouping is a valuable method for breaking down quadratic expressions into simpler components. This process involves rearranging and grouping the terms to find common factors. To employ this technique, we examine the quadratic expression \(j^2 + 9j + 20\) and identify two numbers that multiply to the product of the leading coefficient and the constant term, and also sum to the middle coefficient.For \(j^2 + 9j + 20\), the pair of numbers that work are 4 and 5, as their product is 20 and they sum to 9. We rewrite the middle term, \(9j\), using these numbers: \(4j\) and \(5j\). This alters the expression to \(j^2 + 4j + 5j + 20\).

• Group terms: \((j^2 + 4j) + (5j + 20)\)
• Factor within groups: \(j(j + 4) + 5(j + 4)\)

The expression now has a common factor of \((j+4)\) in both groups, highlighting the ability to simplify further.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. This technique is crucial for solving equations and simplifying expressions. Quadratics can often be factorized into the product of two binomials.In our example, once \(j(j + 4) + 5(j + 4)\) has been factored by grouping, it can be further factorized to reflect \((j + 4)(j + 5)\). Here, \((j + 4)\) is identified as a common factor, which simplifies the expression into the product form.

• This step-by-step factorization process highlights structural relationships within polynomials.
• Polynomial factorization is foundational for solving quadratic equations, as it often leads to simpler equations.

Quadratic Formula

Although the current problem did not require the quadratic formula, understanding it is beneficial for wider applications. The quadratic formula provides a method for finding the roots of any quadratic equation that cannot be easily factorized.The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula derives from completing the square on the general quadratic equation \(ax^2 + bx + c = 0\). It enables solving for \(x\) by determining where the polynomial equals zero.

• The discriminant \(b^2 - 4ac\) within the formula determines the nature of the roots, whether they are real or complex.
• Despite not being used in factoring by grouping, the quadratic formula remains crucial for cases where factorization isn't straightforward.