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Chapter 6: Problem 5

Factor each trinomial. $$ y^{2}+7 y-30 $$

Short Answer

Expert verified
(y + 10)(y - 3)

Step by step solution

01

- Identify coefficients

Identify the coefficients of the trinomial. Here, the trinomial is in the form of \[ay^2 + by + c\] with coefficients: \[a = 1, b = 7, c = -30\]
02

- Find two numbers that multiply to ac and add to b

Multiply the coefficients \(a\) and \(c\) to get \(ac = 1 \times -30 = -30\). Find two numbers that multiply to -30 and add to 7. These numbers are 10 and -3.
03

- Rewrite middle term using the two numbers

Rewrite the middle term (7y) using the two numbers (10 and -3): \[y^2 + 10y - 3y - 30\]
04

- Factor by grouping

Group the terms in pairs and factor each pair: \[y(y + 10) - 3(y +10)\]
05

- Factor out the common binomial

Factor out the common binomial factor \((y + 10)\): \[(y + 10)(y - 3)\]

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Key Concepts

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

Coefficients
A polynomial's coefficients are crucial. They represent the numerical factors associated with each term. For the trinomial \(y^{2} + 7y - 30\), the coefficients are easy to spot. The standard form of a trinomial is \(ay^2 + by + c\). Here:
  • \(a = 1\): This is the coefficient in front of \(y^2\)
  • \(b = 7\): This is the middle term's coefficient
  • \(c = -30\): This is the constant term
These coefficients will help us in breaking down and simplifying the trinomial further. Understanding these values is the first step in factoring the trinomial.
Factoring by Grouping
Factoring by grouping is a method used to simplify trinomials. We break the middle term into two parts that will help in factoring. For our trinomial: \(y^2 + 7y -30\)
  • Find two numbers that multiply to the product of the first and last coefficients (here, 1 and -30). The product is -30.
  • These two numbers should add up to the middle coefficient 'b' (which is 7). For our example, these numbers are 10 and -3.
  • We then rewrite the trinomial by breaking the middle term: \(y^2 + 10y - 3y - 30\).
Grouping helps us to factor the quadratic trinomial in a structured way. It makes it easier to see which terms combine to form a factorable expression.
Binomial Factorization
Binomial factorization involves factoring a polynomial into two binomials (expressions with two terms). For the polynomial \(y^2 + 10y - 3y - 30\):
  • We apply grouping: \[y(y + 10) -3(y + 10)\], grouping them into two pairs.
  • Here, we see a common binomial factor (\(y + 10\)). We factor it out from both groups: \[(y + 10)(y - 3)\].
This shows how we can transform a trinomial into the product of two simpler binomial expressions. The result, \((y + 10)(y - 3)\), is the factored form of our original polynomial.
Restructuring Middle Term
Restructuring the middle term is a critical step. We use the two numbers found earlier that multiply to our original product \(ac\) and add to our middle term coefficient \(b\). For the trinomial \(y^2 + 7y - 30\):
  • We found that 10 and -3 multiply to -30 and add to 7.
  • We rewrite the middle term, \(7y\), as: \(10y -3y\)
  • The trinomial now looks like: \(y^2 + 10y - 3y -30\),
Breaking the middle term into two terms lets us use the grouping method to factorize. This step bridges identifying coefficients and grouping terms. It transforms the trinomial into a format that is easier to factor.

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Most popular questions from this chapter

In some cases, the method of factoring by grouping can be combined with the methods of special factoring discussed in this section. Consider this example. $$ \begin{aligned} 8 x^{3} &+4 x^{2}+27 y^{3}-9 y^{2} \\ &=\left(8 x^{3}+27 y^{3}\right)+\left(4 x^{2}-9 y^{2}\right) \end{aligned} $$ $$ \begin{aligned} &=(2 x+3 y)\left(4 x^{2}-6 x y+9 y^{2}\right)+(2 x+3 y)(2 x-3 y)\\\ &=(2 x+3 y)\left[\left(4 x^{2}-6 x y+9 y^{2}\right)+(2 x-3 y)\right]\\\ &=(2 x+3 y)\left(4 x^{2}-6 x y+9 y^{2}+2 x-3 y\right) \end{aligned} $$ In problems such as this, how we choose to group in the first step is essential to factoring correctly. If we reach a "dead end," then we should group differently and try again. Use the method just described to factor each polynomial. $$ 64 m^{2}-512 m^{3}-81 n^{2}+729 n^{3} $$

If a rock is dropped from a building 576 ft high, then its distance in feet from the ground \(t\) seconds later is a function defined by $$f(t)=-16 t^{2}+576$$ How long after it is dropped will it hit the ground?

Factor each trinomial. $$ -15 a^{2}-70 a+120 $$

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