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Chapter 6: Problem 32

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$r^{2}-15 r-16$$

Short Answer

Expert verified
The factorization of the trinomial \( r^{2}-15r-16 \) is \((r - 16)(r + 1)\).

Step by step solution

01

Identify Factors

The general form of a trinomial is \(ax^{2} + bx + c\). In this case, \(a=1\), \(b=-15\), and \(c=-16\). We search for two numbers that multiply to \(ac = -16\), and add up to \(b = -15\). Those numbers are -16 and 1, because \( -16 * 1 = -16 \) and \( -16 + 1 = -15 \).
02

Factor the Trinomial

According to the identified factors, the trinomial can be factored. The expression becomes: \((r - 16)(r + 1)\), where \(r - 16\) and \(r + 1\) are the factors of the given trinomial.
03

Check the Factorization

Use the FOIL method to check if the result from Step 2 is correct. Multiply the first terms: \( r * r = r^{2} \). Multiply the outside terms: \( r * 1 = r \). Multiply the inside terms: \( -16 * r = -16r \). Multiply the last terms: \( -16 * 1 = -16 \). Add these together: \( r^{2} + r - 16r - 16 = r^{2} - 15r - 16\). This is the original trinomial, so the factorization is correct.

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Key Concepts

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

FOIL Method
The FOIL Method is a handy tool for multiplying binomials. The acronym stands for First, Outside, Inside, Last, which are the pairs of terms you'll multiply together:
  • First: Multiply the first terms of each binomial.
  • Outside: Multiply the outer terms of the two binomials.
  • Inside: Multiply the inner terms.
  • Last: Multiply the last terms in each binomial.
This method simplifies the multiplication process and ensures that no terms are omitted. For instance, when we multiplied \((r - 16)(r + 1)\) using FOIL, it confirmed our factorization was correct because it returned us to the original trinomial \\(r^2 - 15r - 16\). Thus, learning the FOIL method is crucial for verifying polynomial factorization.
Polynomial Factorization
Polynomial Factorization is the process of breaking down a polynomial into simpler components, called factors, that, when multiplied, return the original polynomial. For trinomials like \(r^2 - 15r - 16\), it involves identifying two binomials that multiply to give the original trinomial. Key steps in the process include:
  • Identifying the coefficients \(a\), \(b\), and \(c\) in the general trinomial form \(ax^2 + bx + c\).
  • Finding two numbers that multiply to \(ac\) and add up to \(b\).
  • Using these numbers to split the middle term \(b\) and factor by grouping if necessary.
Once factored, you can verify your result using the FOIL method, ensuring a complete, correct solution.
Algebraic Expressions
Algebraic Expressions include numbers, variables, and operators. They form the building blocks of algebra and offer a way to represent mathematical concepts. For example, the expression \(r^2 - 15r - 16\) is an algebraic expression that includes variables \(r\), coefficients \(-15\) and \(-16\), and an exponent on \(r\). Understanding these components is crucial:
  • Variables are symbols like \(r\) that stand in for unknown values.
  • Coefficients are numbers that multiply the variables. In \(-15r\), \(-15\) is the coefficient.
  • Constants are numbers on their own, in this case, \(-16\).
Recognizing these components helps in manipulating and solving equations, making polynomial factorization possible. Mastering algebraic expressions sets a strong foundation for further mathematical learning.

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Most popular questions from this chapter

Great white sharks have triangular teeth with a height that is 1 centimeter longer than the base. If the area of one tooth is 15 square centimeters, find its base and height.

Factor: \(9 x^{2}-16 .\) (Section 6.4, Example 1)

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(4 y^{2}+20 y+25=0\)

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-12 x+9=(4 x-3)^{2} ;[-5,5,1] \text { by }[0,20,1]$$

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(y(y+8)=16(y-1)\)
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