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Magazine Chapter 5: Problem 57
Factor each trinomial completely. \(15 n^{4}-39 n^{3}+18 n^{2}\)
Short Answer
Expert verified
3n^{2}(n - 2)(5n - 3)
Step by step solution
01
Identify the Greatest Common Factor (GCF)
First, determine the greatest common factor (GCF) of the coefficients 15, -39, and 18, and the smallest power of the variable present in all terms.
02
Extract the GCF
The GCF of 15, -39, and 18 is 3. Since all terms have at least ^{2}, the GCF is 3n^{2}. Factor out the 3n^{2} from the trinomial: 15n^{4} - 39n^{3} + 18n^{2} = 3n^{2}(5n^{2} - 13n + 6).
03
Factor the Quadratic Trinomial
Now, factor the quadratic trinomial 5n^{2} - 13n + 6. Find two numbers whose product is 5 * 6 = 30 and whose sum is -13. These numbers are -10 and -3. Rewrite the quadratic as 5n^{2} - 10n - 3n + 6.
04
Group Terms and Factor by Grouping
Group the terms in pairs and factor each pair: 5n^{2} - 10n - 3n + 6 = (5n^{2} - 10n) + (-3n + 6). Factor out the common factors in each group: 5n(n - 2) - 3(n - 2).
05
Factor the Binomial
Factor out the common binomial factor (n - 2): 5n(n - 2) - 3(n - 2) = (n - 2)(5n - 3).
06
Combine All Factors
Combine the factors with the GCF factored out in step 2: 3n^{2}(5n^{2} - 13n + 6) = 3n^{2}(n - 2)(5n - 3).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Greatest Common Factor
Understanding the Greatest Common Factor (GCF) is crucial in factoring trinomials. The GCF is the largest factor that divides all terms of a polynomial. To find the GCF of a polynomial, follow these steps:
First, identify the GCF of the coefficients (the numerical parts) of the terms. For instance, in the polynomial given, the coefficients are 15, -39, and 18. The GCF of these numbers is 3.
Second, determine the smallest power of the variable that appears in all terms. For example, in the polynomial provided, the variable is n, and the smallest power is n虏.
Combining these two pieces of information, the GCF for the polynomial 15n鈦� - 39n鲁 + 18n虏 is 3n虏. Factoring the GCF out of the polynomial simplifies the expression, making it easier to factor further.
First, identify the GCF of the coefficients (the numerical parts) of the terms. For instance, in the polynomial given, the coefficients are 15, -39, and 18. The GCF of these numbers is 3.
Second, determine the smallest power of the variable that appears in all terms. For example, in the polynomial provided, the variable is n, and the smallest power is n虏.
Combining these two pieces of information, the GCF for the polynomial 15n鈦� - 39n鲁 + 18n虏 is 3n虏. Factoring the GCF out of the polynomial simplifies the expression, making it easier to factor further.
Quadratic Trinomial
A quadratic trinomial is a polynomial of degree 2 with three terms. It takes the form ax虏 + bx + c, where a, b, and c are constants. In the example, 5n虏 - 13n + 6 is a quadratic trinomial.
To factor a quadratic trinomial, identify two numbers that multiply to give the product of the leading coefficient (a) and the constant term (c) and add up to the middle coefficient (b). For 5n虏 - 13n + 6:
- The product of the leading coefficient (5) and the constant term (6) is 30.
- We need two numbers that multiply to 30 and add up to -13.
The numbers we're looking for are -10 and -3. These numbers can help rewrite the middle term, facilitating factoring by grouping.
To factor a quadratic trinomial, identify two numbers that multiply to give the product of the leading coefficient (a) and the constant term (c) and add up to the middle coefficient (b). For 5n虏 - 13n + 6:
- The product of the leading coefficient (5) and the constant term (6) is 30.
- We need two numbers that multiply to 30 and add up to -13.
The numbers we're looking for are -10 and -3. These numbers can help rewrite the middle term, facilitating factoring by grouping.
Factoring by Grouping
Factoring by grouping is an effective method for factoring polynomials, especially when dealing with quadratic trinomials. After rewriting the quadratic trinomial 5n虏 - 13n + 6 as 5n虏 - 10n - 3n + 6, the next step is to group terms:
Group the terms in pairs: (5n虏 - 10n) and (-3n + 6).
- In the first group, factor out the common factor: 5n(n - 2).
- In the second group, factor out the common factor: -3(n - 2).
Now, you will notice that both groups have a common binomial factor, (n - 2). Factoring this out gives: (5n(n - 2) - 3(n - 2)) = (n - 2)(5n - 3).
Group the terms in pairs: (5n虏 - 10n) and (-3n + 6).
- In the first group, factor out the common factor: 5n(n - 2).
- In the second group, factor out the common factor: -3(n - 2).
Now, you will notice that both groups have a common binomial factor, (n - 2). Factoring this out gives: (5n(n - 2) - 3(n - 2)) = (n - 2)(5n - 3).
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of simpler polynomials. The process begins with factoring out the GCF and continues through factoring by grouping or other methods.
In our example, after factoring out the GCF (3n虏), the remaining polynomial is 5n虏 - 13n + 6. Using the factoring by grouping technique, this polynomial is factored as (n - 2)(5n - 3).
Putting it all together, the entire factorization of 15n鈦� - 39n鲁 + 18n虏 is: 3n虏(n - 2)(5n - 3).
Polynomial factorization helps in simplifying problems and solving equations by breaking down complex polynomials into more manageable parts.
In our example, after factoring out the GCF (3n虏), the remaining polynomial is 5n虏 - 13n + 6. Using the factoring by grouping technique, this polynomial is factored as (n - 2)(5n - 3).
Putting it all together, the entire factorization of 15n鈦� - 39n鲁 + 18n虏 is: 3n虏(n - 2)(5n - 3).
Polynomial factorization helps in simplifying problems and solving equations by breaking down complex polynomials into more manageable parts.
