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Chapter 5: Problem 10

The following exercises are of mixed variety. Factor each polynomial. $$ k^{2}-6 k-16 $$

Short Answer

Expert verified
The factored form of \( k^2 - 6k - 16 \) is \( (k - 8)(k + 2) \).

Step by step solution

01

- Identify coefficients

Identify the coefficients in the quadratic expression. The expression is in the standard form of a quadratic equation, \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = -6 \), and \( c = -16 \).
02

- Find two numbers that multiply and sum to specific values

Find two numbers that multiply to give the product of \( a \) and \( c \) (which is \( 1 \times -16 = -16 \)), and add to give \( b \) (which is -6). These numbers are -8 and 2 since \(-8 \times 2 = -16\) and \(-8 + 2 = -6\).
03

- Split the middle term

Rewrite the quadratic expression by splitting the middle term \( -6k \) into two terms using the numbers identified: \( k^2 - 8k + 2k - 16 \).
04

- Factor by grouping

Group the terms in pairs and factor each pair: \( (k^2 - 8k) + (2k - 16) \). Factor out the common factors from each pair: \( k(k - 8) + 2(k - 8) \).
05

- Factor out the common binomial

Factor out the common binomial factor \( k - 8 \): \( (k - 8)(k + 2) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

polynomial factoring
Polynomial factoring is a key skill in algebra that involves breaking down a polynomial into simpler polynomials that, when multiplied, give back the original polynomial. This process is essential for solving polynomial equations and simplifying expressions.

For quadratic expressions, factoring can help us find the roots of the equation or understand its properties. In the given exercise, we factor the quadratic polynomial into the product of two binomials.
quadratic expressions
Quadratic expressions are polynomials where the highest exponent of the variable is 2. The general form is:
\(a x^2 + b x + c\),
where \(a\), \(b\), and \(c\) are constants.

In the provided exercise, the quadratic expression is:
\(k^2 - 6k - 16\).

Here, \(a = 1\), \(b = -6\), and \(c = -16\). By identifying these coefficients, we can proceed to factor the quadratic expression.
algebraic manipulation
Algebraic manipulation refers to techniques used to simplify or solve algebraic expressions. In the solved exercise, we use algebraic manipulation to break down the quadratic expression:
1. Identify coefficients \(a = 1\), \(b = -6\), \(c = -16\).
2. Find numbers that multiply to \(a \times c = -16\) and add to \(b = -6\).
3. Split the middle term: \(-6k\) becomes \(-8k + 2k\).

These steps help us reframe the expression for easier factoring.
factoring by grouping
Factoring by grouping is a method used to factor polynomials, especially helpful for quadratic expressions. Here's how it's applied in the exercise:
1. Rewrite the quadratic expression: \(k^2 - 8k + 2k - 16\).
2. Group terms: \((k^2 - 8k) + (2k - 16)\).
3. Factor out common terms from each group: \(k(k - 8) + 2(k - 8)\).
4. Identify the common binomial and factor it out: \((k - 8)(k + 2)\).

This method simplifies the expression into a product of binomials.

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Most popular questions from this chapter

Find the greatest common factor for each list of terms. $$ 10 m^{2} n^{2}, 25 m n^{3}, 50 m^{2} n $$

Factor each polynomial. $$ 25 c^{2}-20 c+4-d^{2} $$

A square mirror has sides measuring \(2 \mathrm{ft}\) less than the sides of a square painting. If the difference between their areas is \(32 \mathrm{ft}^{2},\) find the lengths of the sides of the mirror and the painting.

A box with no top is to be constructed from a piece of cardboard whose length measures 6 in. more than its width. The box is to be formed by cutting squares that measure 2 in. on each side from the four corners and then folding up the sides. If the volume of the box will be \(110 \mathrm{in} .^{3}\), what are the dimensions of the piece of cardboard?

Factor each trinomial. \(8 k^{2}+34 k+35\)
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