Problem 39 - Beginning and Intermediate Algebra

Question

Factor completely by first taking out \(-1\) and then by factoring the trinomial, if possible. Check your answer. $$-k^{2}+11 k-28$$

Step-by-Step Solution

Answer

The completely factored expression for the given quadratic trinomial is \(-1(k - 4)(k - 7)\).

Step 1: Take out -1

Begin by taking out -1 from the expression: $$-k^2 + 11k - 28 = -1(k^2 - 11k + 28)$$

Step 2: Factor the trinomial

Now, we need to factor the quadratic expression inside the parentheses, if possible. To factor the quadratic expression, we need to find two numbers whose multiplication equals the product of the first and last terms, and the sum equals the middle term: - Product: \((k^2)(28) = 28k^2\) - Sum: \(-11k\) The two numbers required for factoring the trinomial are -4 and -7.

Step 3: Rewrite the trinomial

Write down the trinomial as the product of two binomials using the two numbers found in the previous step: $$k^2 - 11k + 28 = (k - 4)(k - 7)$$ Now, substitute the factored trinomial back into the expression obtained in Step 1: $$-1(k^2 - 11k + 28) = -1(k - 4)(k - 7)$$

Step 4: Check the answer

To check the answer, expand the factored expression and see if it matches the original expression: \((-1)(k-4)(k-7) = (k-4)(-k+7)\) Expanding the factored expression, we get: \((k-4)(-k+7) = -k^2 + 7k + 4k - 28 = -k^2 + 11k - 28\) Since the expanded expression matches the original expression, our factoring is correct.

Step 5: Solution:

The completely factored expression for the given quadratic trinomial is $$-1(k - 4)(k - 7)$$.

Key Concepts

Trinomial

A trinomial is a polynomial with exactly three terms. In the context of quadratic expressions, a typical trinomial might look like this: \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are coefficients, with \(a\) being non-zero. Understanding how to work with trinomials is crucial when it comes to polynomial expressions, as they often need to be simplified or factored.
Trinomials can often be factored into the product of two binomial expressions. This process involves finding two numbers that will multiply to give the product of the first and third coefficients and also add up to the second coefficient. These numbers are key to breaking down the trinomial into simpler, more manageable parts.

• Recognize a trinomial as having three terms.
• Look for a structure: \(ax^2 + bx + c\).
• Know that factoring involves breaking it into two binomial expressions.

Whether used in solving quadratic equations or simplifying expressions, mastering trinomials helps in tackling a range of mathematical problems.

Factoring by Grouping

Factoring by grouping is a technique used when dealing with polynomials, especially when the terms do not immediately suggest how they should be factored. This method works well for quadratic expressions and involves splitting the middle term into two terms that allow the expression to be grouped and factored in parts.
For example, if you have a polynomial expression such as \(ax^2 + bx + c\), you might begin by finding two numbers that both multiply to the product of \(a\times c\) and sum to \(b\). Once you have these numbers, you can rewrite the polynomial, splitting the middle term accordingly and then factor by grouping.
When applying this to factor the expression \(x^2 - 11x + 28\), you look for two numbers that multiply to \(28x^2\) and add to \(-11x\). In this case, \(-4x\) and \(-7x\) are used to split \(-11x\) before grouping and factoring further.

• Identify terms to group separately.
• Ensure they allow for common factorization.
• Reassess the expression once grouped and factored.

This approach is particularly intuitive when a direct factorization approach does not immediately present itself.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest power of the variable is 2. The standard form is \(ax^2 + bx + c\), where \(a\) cannot be 0. These expressions are fundamental in algebra because they appear in various mathematical problems, including equations, functions, and graphs.
An important task is understanding how to factor or solve these expressions. When factoring, the aim is to transform the expression into a product of simpler expressions (usually two binomials). This can be advantageous when solving quadratic equations because it leads to finding the roots or solutions.

• Recognize the quadratic form: \(ax^2 + bx + c\).
• Useful for finding the expression's roots.
• Algebraic methods such as completing the square and the quadratic formula can solve these expressions.

Factoring quadratic expressions is sometimes straightforward, yet other techniques may be necessary, particularly if the factoring approach seems overly complex.

Polynomials

Polynomials are algebraic expressions that consist of variables, coefficients, and exponents combined through addition, subtraction, and multiplication. These expressions are broad and include terms with varying powers, with no negative or fractional exponents.
Polynomials are categorized by their degree, which is determined by the highest exponent of the variable in the expression. Trinomials, binomials, and even monomials are specific types of polynomials characterized by the number of terms they contain.
Factoring polynomials, including trinomials, is a common task in algebraic manipulation, simplifying expressions, and solving equations. The process involves finding simpler polynomials that multiply to give the original polynomial.

• Understand polynomial structure and terminology.
• Recognize common types: monomials, binomials, trinomials.
• Use factoring to simplify and solve polynomial equations.

Mastering polynomials forms a foundational aspect of higher-level mathematics and is crucial in various problem-solving scenarios.