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Magazine Chapter 3: Problem 28
Factor completely each of the polynomials and indicate any that are not factorable using integers. $$4 n^{2}-19 n+21$$
Short Answer
Expert verified
The polynomial \(4n^2 - 19n + 21\) factors to \((n - 3)(4n - 7)\).
Step by step solution
01
Identify the Polynomial Structure
The given polynomial is a quadratic in the form \( ax^2 + bx + c \), where \( a = 4 \), \( b = -19 \), and \( c = 21 \). Our task is to factor this expression completely.
02
Search for Integer Factors of ac
Calculate \( ac = 4 \times 21 = 84 \). We need to find two integers whose product is 84 and whose sum is \(-19\).
03
List Integer Pairs
Consider pairs of integers that multiply to 84: \( (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), (7, 12) \). We are looking for such a pair where the sum is \(-19\).
04
Identify Suitable Pair
Check the pair \( (-7, -12) \), as it meets both criteria: the product \((-7) \times (-12) = 84\) and the sum \(-7 + (-12) = -19\).
05
Rewrite the Middle Term
Rewrite the middle term \(-19n\) of the quadratic as \(-7n - 12n\). The expression becomes \(4n^2 - 7n - 12n + 21\).
06
Factor by Grouping
Group the terms: \((4n^2 - 7n) + (-12n + 21)\). Factor each group: \(n(4n - 7) - 3(4n - 7)\).
07
Extract the Common Factor
Notice \((4n - 7)\) is a common factor: \((n - 3)(4n - 7)\). Thus, the factored form of the polynomial is \((n - 3)(4n - 7)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Expressions
Polynomial expressions are mathematical phrases made up of variables, coefficients, and exponents. A polynomial involves operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A basic form of a polynomial expression is \( ax^n + bx^{n-1} + ... + k \), where \( a, b, ... \), and \( k \) are constants.
- Quadratic polynomials are a specific type of polynomial expression that can be written in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).
- The degree of a polynomial is the highest power of the variable in the expression. For a quadratic polynomial, the degree is \(2\).
- Quadratic polynomials often need to be factored into simple binomial expressions to find solutions or simplify expressions further.
Integer Factorization
Integer factorization involves writing an integer or a polynomial as a product of smaller, simpler terms. When dealing with polynomials, especially quadratics, the coefficients are often integers, and we aim to express these polynomials as a product of binomials with integer coefficients.
- The integer factorization of a polynomial means breaking it down into factors that multiply to give the original polynomial.
- To start, identify the coefficient \(ac\) from the quadratic polynomial \(ax^2 + bx + c\). Then, search for two integers that multiply to \(ac\) and add up to \(b\).
- This process can be likened to solving a puzzle where you're finding compatible pairs that meet defined criteria.
- Integer factorization simplifies a polynomial, making it easier to manipulate, solve equations, or integrate into more complex math problems.
Factor by Grouping
Factor by grouping is a technique used when a polynomial has four or more terms. In terms of a quadratic polynomial, factor by grouping is particularly applied after breaking down the middle term.
- Begin by rewriting the middle term of the quadratic expression, ensuring the sum of the new terms equals the original middle term. In our exercise, \(-19n\) was rewritten as \(-7n - 12n\).
- Arrange the expression into groups of terms. Each group should have a common factor so you can easily extract it.
- The objective here is to factor out the greatest common factor from each group. After grouping, such as \((4n^2 - 7n) + (-12n + 21)\), factor out \(n\) from the first group and \(-3\) from the second, resulting in \(n(4n - 7) - 3(4n - 7)\).
- Notice the common binomial in each group, which allows us to further factor the expression into \((n - 3)(4n - 7)\).
- All terms should multiply back to the original polynomial, verifying your factorization.
