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Chapter 5: Problem 70

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

Short Answer

Expert verified
(x+y)(n+5)(n-4)

Step by step solution

01

- Identify the common factor

Observe that \( (x+y) \) is a common factor in every term of the polynomial. The polynomial is \( (x+y) n^{2}+(x+y) n-20(x+y) \).
02

- Factor out the common term

Factor out \( (x+y) \) from each term in the polynomial. This gives \( (x+y) \) times the remaining terms inside the parentheses: \[ (x+y)(n^{2}+n-20) \].
03

- Factor the quadratic expression

Next, factor the quadratic expression \( n^{2}+n-20 \). Find two numbers that multiply to -20 and add to 1. These numbers are 5 and -4. Therefore, \ (n+5)(n-4) \.
04

- Combine the factors

Combine the two factored parts to get the complete factored form of the original polynomial: \[ (x+y)(n+5)(n-4) \].

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Key Concepts

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

Common Factor
In algebra, a common factor refers to a number or expression that divides each term in a polynomial without leaving a remainder. In the given polynomial \((x+y) n^{2}+(x+y) n-20(x+y)\), we can observe that the expression \((x+y)\) appears in every term. This makes \((x+y)\) the common factor.

Factoring out the common factor simplifies the polynomial by grouping similar terms together. Here, we can factor out \((x+y)\), leaving us with: \[(x+y) (n^{2}+n-20)\].
  • Always look for terms that repeat.
  • Factor them out to simplify expressions.
Quadratic Expression
A quadratic expression is a polynomial of degree 2, typically in the form of \(ax^{2} + bx + c\). In our exercise, the quadratic expression within the parentheses is \(n^{2}+n-20\).

To factor a quadratic expression, we need to find two numbers that multiply to the constant term (in this case, -20) and add up to the coefficient of the linear term (1 here).
  • These numbers are 5 and -4 because \(5 \times -4 = -20\) and \(5 + (-4) = 1\).
  • Thus, we can write \(n^{2}+n-20 = (n+5)(n-4)\).
Factored Form
The factored form of a polynomial is a way of expressing the polynomial as a product of its factors. In the original exercise, after factoring out the common factor and breaking down the quadratic expression, we combine them to get the final factored form.

Starting from: \[(x+y) n^{2}+(x+y) n-20(x+y)\]
Turn it into: \[(x+y)(n^{2}+n-20)\]
Then factor \(n^{2}+n-20\) as \((n+5)(n-4)\)
So the completely factored form is: \[(x+y)(n+5)(n-4)\]
This form is very useful for solving equations and understanding the roots of the polynomial.
  • First factor out the common term.
  • Then factor any simpler quadratic expressions.

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Most popular questions from this chapter

Factor completely. \(a^{2}-a b-12 b^{2}\)

Find the value of the indicated variable. Find \(b\) so that \(100 a^{2}+b a+9\) factors as \((10 a+3)^{2}\).

Factor each trinomial completely. See Examples 1鈥�7. ( Hint: In Exercises 55鈥�58, first write the trinomial in descending powers and then factor.) $$ 15 n^{4}-39 n^{3}+18 n^{2} $$

Students often become confused as to how to handle a constant, such as 2 in the equation \(2 x(3 x-4)=0 .\) How would you explain to someone how to solve this equation and how to handle the constant \(2 ?\)

Find the value of the indicated variable. Find \(a\) so that \(a y^{2}-12 y+4\) factors as \((3 y-2)^{2}\).
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