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Chapter 5: Problem 37

Factor completely. If the polynomial cannot be factored, write prime. \(r^{2}-r-30\)

Short Answer

Expert verified
The polynomial factors as \((r - 6)(r + 5)\).

Step by step solution

01

Identify the terms

Recognize that the polynomial is a quadratic expression in the form of \(ax^2 + bx + c\) where \(a = 1\), \(b = -1\), and \(c = -30\).
02

Find two numbers that multiply to \(c\) and add to \(b\)

Look for two numbers that multiply to \(-30\) (the constant term c) and add up to \(-1\) (the coefficient of r). These numbers are \( -6\) and \( 5\) because \((-6) \times 5 = -30\) and \((-6) + 5 = -1\).
03

Rewrite the middle term using these two numbers

Rewrite the expression \(r^2 - r - 30\) using \(-6\) and \(5\): \(r^2 - 6r + 5r - 30\).
04

Factor by grouping

Group the terms in pairs and factor out the common factors in each group: \([r^2 - 6r] + [5r - 30]\). Factor out \(r\) from the first group and \(5\) from the second group: \[r(r - 6) + 5(r - 6)\].
05

Factor out the common binomial

Notice that \((r - 6)\) is common in both terms, so factor out the common binomial factor: \((r - 6)(r + 5)\).

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Key Concepts

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

quadratic polynomials
A quadratic polynomial is a type of polynomial where the highest degree of the variable is 2. This means that the variable is squared. For instance, in the expression \(r^2 - r - 30\), the highest power of the variable \r\ is 2. Quadratic polynomials typically have the form \(ax^2 + bx + c\), where \a\, \b\, and \c\ are constants, and \a \eq 0\. In our exercise, the constants are: \a = 1\, \b = -1\, and \c = -30\.

Understanding the structure of quadratic polynomials helps in recognizing their properties and patterns, which is essential when you start factoring or solving them.
factoring by grouping
Factoring by grouping is a method used to factor polynomials, especially when they have four terms. This method groups terms with common factors, making it easier to factor completely.

In our exercise, we start with \(r^2 - r - 30\). To use factoring by grouping, we:
  • Find two numbers that multiply to \c\ (the constant term) and add to \b\ (the coefficient of the middle term). Here, these numbers are \ -6 \ and \ 5\. Their product is \ -30\ and their sum is \ -1\.
  • Rewrite the polynomial using these two numbers: \(r^2 - 6r + 5r - 30\).
  • Group the terms into pairs: \[(r^2 - 6r) + (5r - 30)ewline\]\, factor each group separately: \ [r(r - 6) + 5(r - 6)]\.
  • Finally, factor out the common binomial factor: \( (r - 6)(r + 5)\).
solving polynomials
Solving a polynomial involves finding the values of the variable that make the polynomial equal to zero. For a factored quadratic polynomial like \( (r - 6)(r + 5) \), we set each factor equal to zero and solve for the variable.

This means solving: \ r - 6 = 0 \ and \ r + 5 = 0 \, which gives us the solutions \ r = 6 \ and \ r = -5\. These solutions are called the roots of the polynomial.

Understanding how to solve polynomials is crucial, as it helps in finding the roots, which can be essential for graphing and for solving real-world problems related to the polynomial.

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Most popular questions from this chapter

Solve each equation, and check your solutions. $$(x+3)^{2}-(2 x-1)^{2}=0$$

Find product. \((2 y-7)(y+4)\)

Students often have difficulty when factoring by grouping because they are not able to tell when the polynomial is completely factored. For example, $$5 y(2 x-3)+8 t(2 x-3)$$ is not in factored form, because it is the sum of two terms: \(5 y(2 x-3)\) and \(8 t(2 x-3)\) However, because \(2 x-3\) is a common factor of these two terms, the expression can now be factored. $$(2 x-3)(5 y+8 t)$$ The factored form is a product of two factors: \(2 x-3\) and \(5 y+8 t\) Determine whether each expression is in factored form or is not in factored form. If it is not in factored form, factor it if possible. $$ (3 r+7)(5 x-1) $$

Apply the special factoring rules of this section to factor each binomial or trinomial. $$ p^{2}-\frac{1}{9} $$

Factor polynomial. \((x+y) n^{2}+(x+y) n-20(x+y)\)
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