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Chapter 5: Problem 37

Factor each trinomial completely. \(20 y^{2}-39 y-11\)

Short Answer

Expert verified
The trinomial factors to \[(4y + 1)(5y - 11)\]

Step by step solution

01

- Identify and Check the Trinomial

The given trinomial is \[20y^2 - 39y - 11\]
02

- Multiply Coefficients of the First and Last Terms

Multiply the coefficients of the first term (20) and the last term (-11): \[20 \times (-11) = -220\]
03

- Find Two Numbers that Multiply to -220 and Add to -39

Identify two numbers that multiply to -220 and add up to -39. These numbers are -44 and 5 because: \[-44 \times 5 = -220\] and \[-44 + 5 = -39\]
04

- Rewrite the Middle Term

Rewrite the middle term, \[-39y\], using the two numbers found in Step 3:\[20y^2 - 44y + 5y - 11\]
05

- Factor by Grouping

Group the terms into pairs and factor out the greatest common factor from each pair:\[20y^2 - 44y + 5y - 11 = 4y(5y - 11) + 1(5y - 11)\]
06

- Factor the Common Binomial Factor

Factor out the common binomial factor (5y - 11):\[4y(5y - 11) + 1(5y - 11) = (4y + 1)(5y - 11)\]

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Key Concepts

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

trinomial factorization
Trinomial factorization is an essential skill in algebra. It involves breaking down a trinomial (a polynomial with three terms) into simpler binomial factors. This process is useful in solving quadratic equations and simplifies complex algebraic expressions. In our example, we had to factor the trinomial \[20y^2 - 39y - 11\].

The essential steps include:
  • Identifying the trinomial and ensuring it's in the standard form, \(ax^2 + bx + c\).
  • Multiplying the coefficient of the first term (\

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Most popular questions from this chapter

Solve each problem. Find three consecutive even integers such that the sum of the squares of the first and second integers is equal to the square of the third integer.

Apply the special factoring rules of this section to factor each polynomial. \(m^{2}+\frac{2}{3} m+\frac{1}{9}\)

Solve each problem. The length of a rectangle is 4 in. longer than its width. The diagonal is 8 in. longer than the width. Find the length, width, and diagonal measures of the rectangle.

Factor each trinomial. \(24 x^{4}+17 x^{2}-20\)

Factor each trinomial. \(36 x^{4}+59 x^{2} y+24 y^{2}\)
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