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Chapter 8: Problem 6

Write each function in factored form, if possible, using integer coefficients. a. \(f(x)=x^{2}+2 x-15\) d. \(k(p)=p^{2}+5 p+7\) b. \(g(x)=x^{2}-6 x+9\) e. \(l(s)=5 s^{2}-37 s-24\) c. \(h(a)=a^{2}+6 a-16\) f. \(m(t)=5 t^{2}+t+1\)

Short Answer

Expert verified
a. (x + 5)(x - 3), b. (x - 3)(x - 3), c. (a + 8)(a - 2), d. Cannot be factored, e. (5s + 3)(s - 8), f. Cannot be factored.

Step by step solution

01

Identify the function to factor

Start with the function provided. For part a, the function given is: f(x) = x^2 + 2x - 15
02

Find factors of the constant term

Identify the factors of the constant term (-15) that add up to the coefficient of the linear term (2). The pairs of factors for -15 are (-1, 15), (1, -15), (-3, 5), and (3, -5). The pair that adds up to 2 are (5 and -3).
03

Write the factored form

Using the factors identified, write the function in its factored form: f(x) = (x + 5)(x - 3)
04

Confirm the factors for each remaining function

Repeat Steps 1-3 for each function. b. g(x): Identify factors of 9 that add up to -6. The factors are (3 and 3), so: g(x) = (x - 3)(x - 3)c. h(a): Identify factors of -16 that add up to 6. The factors are (8 and -2), so: h(a) = (a + 8)(a - 2)
05

Factor functions with non-integer solutions (if applicable)

d. For the equation k(p) = p^2 + 5p + 7, there are no integer factors of 7 that add up to 5. Thus, k(p) cannot be factored using integer coefficients.e. l(s): Identify factors of -120 that add up to -37. The correct pair is (3 and -40), so: l(s) = (5s + 3)(s - 8)f. For the equation m(t) = 5t^2 + t + 1, there are no integer factors of 5 that add up to 1. Thus, m(t) cannot be factored using integer coefficients.

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Key Concepts

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

Integer Coefficients
In algebra, integer coefficients are coefficients that are whole numbers, either positive or negative, including zero. When we say a polynomial has integer coefficients, we mean that all the coefficients in the polynomial are integers. For example, in the polynomial \(f(x) = x^2 + 2x - 15\), the coefficients 1, 2, and -15 are all integers. This is important in polynomial factorization because we often look for factors that also have integer values, making calculations simpler and more straightforward.
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors. Factors can be numbers, variables, or other polynomials. For instance, factorizing the quadratic polynomial \(f(x) = x^2 + 2x - 15\) involves writing it as a product of two binomials: \( (x + 5)(x - 3) \). This means that multiplying \( (x + 5) \) and \( (x - 3) \) will return the original polynomial. To factor a polynomial:
  • Identify pairs of factors of the constant term that add up to the coefficient of the linear term.
  • Use these pairs to rewrite the polynomial in factored form.
Not all polynomials can be factored using integer coefficients. Some may not have integer solutions at all.
Quadratic Functions
A quadratic function is any function that can be described by an equation of the form \(ax^2 + bx + c\), where a, b, and c are constants. The simplest quadratic function is \(y = x^2\). Quadratics are characterized by their U-shaped graphs called parabolas. In this exercise, the given functions are examples of quadratic equations like \(f(x)= x^2 + 2x - 15\). To factor such quadratic functions:
  • Identify the coefficients a, b, and c.
  • Find two numbers that multiply to ac and add to b.
  • Rewrite and group terms to factor by grouping.
Understanding quadratic functions and their properties is essential for solving algebraic equations effectively.
Algebraic Equations
Algebraic equations are mathematical statements that show the equality of two expressions. They can involve variables, constants, and operational symbols. For example, \(x^2 + 2x - 15 = 0\) is an algebraic equation derived from the polynomial function \(f(x) = x^2 + 2x - 15\). Solving these equations involves finding the values of the variables that make the equation true. In the context of polynomial factorization:
  • Set the factored form of the polynomial equal to zero.
  • Solve for the variable to find the roots or solutions of the equation.
Factoring is a critical skill in solving algebraic equations, especially those involving polynomials and quadratic functions.

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Most popular questions from this chapter

If we know the radius and depth of a parabolic reflector, we also know where the focus is. a. Find a generic formula for the focal length \(f\) of a parabolic reflector expressed in terms of its radius \(R\) and depth \(D\). The focal length \(\left|\frac{1}{4 a}\right|\) is the distance between the vertex and the focal point. Assume \(a>0\). b. Under what conditions does \(f=D ?\)

a. In economics, revenue \(R\) is defined as the amount of money derived from the sale of a product and is equal to the number \(x\) of units sold times the selling price \(p\) of each unit. What is the equation for revenue? b. If the selling price is given by the equation \(p=-\frac{1}{10} x+20,\) express revenue \(R\) as a function of the number \(x\) of units sold. c. Using technology, plot the function and estimate the number of units that need to be sold to achieve maximum revenue. Then estimate the maximum revenue.

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