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Magazine Chapter 6: Problem 109
In the following exercises, factor completely using trial and error. $$ -30 q^{3}-140 q^{2}-80 q $$
Short Answer
Expert verified
-10q (q + 4) (3q + 2)
Step by step solution
01
- Identify common factors
Examine the given polynomial ewline -30q^{3} - 140q^{2} - 80q ewline Identify the common factors in all the terms. In this case, each term includes a factor of -10q.
02
- Factor out the greatest common factor (GCF)
Factor out the greatest common factor (-10q) from each term in the polynomial: ewline -10q(3q^{2} + 14q + 8)
03
- Factor the remaining polynomial
We'll now factor the remaining polynomial (3q^{2} + 14q + 8). We need to find two numbers that multiply to 3*8 (24) and add up to 14. These numbers are 12 and 2.ewlineRewrite the middle term using these numbers: ewline 3q^{2} + 12q + 2q + 8.
04
- Group terms and factor by grouping
Group the terms in pairs and factor out the common factors from each pair: ewline (3q^{2} + 12q) + (2q + 8) ewline = 3q(q + 4) + 2(q + 4).
05
- Factor out the common binomial
Factor out the common binomial factor (q + 4): ewline = (q + 4)(3q + 2).
06
- Combine all the factors
Combine the factor from the greatest common factor step and the factored polynomial: ewline -10q (q + 4) (3q + 2)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
greatest common factor
To begin factoring a polynomial, identifying the greatest common factor (GCF) is crucial. The GCF is the largest factor that divides each term in the polynomial. Finding the GCF involves:
Notice that each term shares a factor of \( -10q \):
- Listing the factors of each term
- Identifying the common factors
- Choosing the highest common factor among them
Notice that each term shares a factor of \( -10q \):
- -30q鲁 = -10q * 3q虏
- -140q虏 = -10q * 14q
- -80q = -10q * 8
trial and error method
The trial and error method, also known as guessing and checking, is useful for factoring polynomials. This method involves finding pairs of numbers that:
We need pairs that multiply to 24 (the product of 3 and 8) and sum to 14. These numbers are 12 and 2.
The middle term \(14q \) can be written as \(12q + 2q \), allowing us to rewrite the polynomial: \(3q^{2} + 12q + 2q + 8 \). This strategic writing helps group and factor by grouping.
- Multiply to give the product of the leading coefficient and the constant term
- Add to give the middle coefficient
We need pairs that multiply to 24 (the product of 3 and 8) and sum to 14. These numbers are 12 and 2.
The middle term \(14q \) can be written as \(12q + 2q \), allowing us to rewrite the polynomial: \(3q^{2} + 12q + 2q + 8 \). This strategic writing helps group and factor by grouping.
factoring by grouping
Factoring by grouping is a method where we group the terms in pairs to help factor a polynomial. For the polynomial \( 3q^{2} + 12q + 2q + 8 \), follow these steps:
\( (q + 4)(3q + 2) \). By grouping, we successfully factor the polynomial into simpler binomials.
- Group terms in pairs: \((3q^{2} + 12q) + (2q + 8) \)
- Factor out the common factors from each pair: \( 3q(q + 4) + 2(q + 4) \)
- Notice the common binomial \((q + 4)\) in both pairs
\( (q + 4)(3q + 2) \). By grouping, we successfully factor the polynomial into simpler binomials.
polynomial expressions
Polynomial expressions consist of variables and coefficients combined using addition, subtraction, and multiplication. For example, in \( -30q^{3} - 140q^{2} - 80q \), the expression includes:
- Terms: \(-30q^{3}\), \(-140q^{2}\), and \(-80q\)
- Coefficients: -30, -140, and -80
- Variables: \(q\)
- Exponents: 3, 2, and 1
