Problem 15 - Beginning and Intermediate Algebra

Question

Factor completely. $$9 k^{2}-24 k+16$$

Step-by-Step Solution

Answer

The completely factored expression is: \( (3k - 2)(3k - 8) \).

Step 1: Identify a common factor for all coefficients

The given expression is \(9k^2 - 24k + 16\). First, let's check if there's a common factor for all the coefficients: - The factors of 9 are 1, 3, and 9 - The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 - The factors of 16 are 1, 2, 4, 8, and 16 We can see that the common factor for all coefficients is 1. However, since dividing by 1 does not change the expression, we should look for another common factor. In this case, 3 is also a common factor for 9 and 24.

Step 2: Factor out the common factor

Now let's factor the expression by taking out the common factor 3: \(9k^2 - 24k + 16 = 3(3k^2 - 8k + 5\frac{1}{3})\) However, since the remaining expression still has a fraction, we cannot consider it factored completely. So, in this case, unfortunately, the common factor technique does not lead us to the complete factorization. Instead, we can try factoring the quadratic expression as a product of two binomials.

Step 3: Factor the quadratic expression

To factor the given quadratic expression completely, we can try writing it as a product of two binomials: \(9k^2 - 24k + 16 = (ak + b)(ck + d)\) Now we need to find a, b, c, and d such that: - The product of the first terms is equal to the quadratic term: \(ac = 9\) - The product of the constant terms is equal to the constant term in the expression: \(bd = 16\) - The sum of the cross-products is equal to the linear term: \(ad + bc = -24\)

Step 4: Determine the values for a, b, c, and d

Possible values for the pairs (a, c), (b, d) that fulfill the necessary conditions are: (a, c): (1, 9) or (3, 3) (b, d): (1, 16), (2, 8), or (4, 4) Testing the possible combinations, we find that the combination (a, c) = (3, 3) and (b, d) = (2, 8) satisfies all the requirements: - \(ac = 3 \cdot 3 = 9\) - \(bd = 2 \cdot 8 = 16\) - \(ad + bc = 3 \cdot 8 + 3 \cdot 2 = 24 \, (then, with negative signs)\) Throwing in the negative signs, our correct values for a, b, c, and d are a = 3, b = -2, c = 3, and d = -8: \(9k^2 - 24k + 16 = (3k - 2)(3k - 8)\)

Step 5: Write the completely factored expression

Now that we have found the correct values for a, b, c, and d, we can write the completely factored expression: \(9k^2 - 24k + 16 = (3k - 2)(3k - 8)\)

Key Concepts

Quadratic Expression

A quadratic expression is a type of polynomial that includes a variable raised to the power of 2. The general form is given as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\). In these expressions, \(x^2\) is known as the quadratic term, \(x\) is the linear term, and the constant is just \(c\). These types of expressions can be found in many areas of algebra and mathematics as they define parabolas in graphs.

• The quadratic term \(ax^2\) affects the width and direction (up or down) of the parabola.
• The linear term \(bx\) influences the axis of symmetry and the position of the vertex along the x-axis.
• The constant term \(c\) affects the y-intercept.

Quadratic expressions can sometimes be tricky to work with, thus factoring plays a crucial role in simplifying these expressions to find their roots or zeros. Understanding the structure of a quadratic is vital before we can move on to factor them effectively.

Factoring Binomials

Factoring binomials involves rewriting a quadratic expression as a product of two simpler expressions, called binomials, usually of the form \((px + q)(rx + s)\). The goal of factoring is to identify pairs of terms that, when multiplied out, recreate the original quadratic expression. This process requires some creative thinking and problem solving, as you need to:

• Find two numbers of products that match the quadratic term coefficient \(a\) and the constant term \(c\).
• Check that the sum of the products matches the linear coefficient \(b\).

For the expression \(9k^2 - 24k + 16\), it was successfully factored into \((3k - 2)(3k - 8)\). The factors were found by trial and error with the multiplication factors for trinomial terms.Factoring binomials is an essential skill in solving quadratic equations, finding intersection points, and solving system inequalities geometrically using graphs.

Common Factors

Finding common factors is an initial step before factoring any quadratic expression. It involves identifying if there are any factors common across all terms in a polynomial. A common factor is a number that can divide each term of an expression evenly without leaving a remainder.

• Start by listing the factors of each coefficient.
• Identify the largest factor that appears in each list.

For the expression \(9k^2 - 24k + 16\), the common factor for 9, 24, and 16 was initially considered as 1. However, when considering the complete factorization, ensuring no fractions arise is crucial, and factoring directly from the polynomial's structure might be more efficient.In cases where the common factor significantly simplifies the expression, factoring it out first makes subsequent factoring steps more straightforward.

Complete Factorization

Complete factorization means expressing the polynomial entirely as a product of its factors until it cannot be factored any further using integers. This process results in a set of factors that are often simpler to understand and manipulate than the original expression.

• Begin by identifying and factoring out any common factors.
• Progress by breaking down the expression through trial and error or using techniques like grouping.
• Confirm the factorization by multiplying the factors to ensure the original expression is reproduced.

The expression \(9k^2 - 24k + 16\) was completely factored into \((3k - 2)(3k - 8)\) in our solution. Achieving complete factorization makes finding solutions to equations more manageable and provides a clearer insight into the behavior of the polynomial.