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Chapter 6: Problem 78

Factor by grouping. \(4 x^{3}+3 x^{2} y+4 x y^{2}+3 y^{3}\)

Short Answer

Expert verified
The polynomial factors to (4x + 3y)(x^2 + y^2).

Step by step solution

01

Identify Grouping Pairs

First, we need to identify pairs of terms that can be grouped together. For the given polynomial, we can group them as follows: (4x^3 + 3x^2y) + (4xy^2 + 3y^3).
02

Factor Common Factors from Each Group

Next, we factor out the greatest common factor (GCF) from each pair of grouped terms. For the first pair (4x^3 + 3x^2y), the GCF is x^2. For the second pair (4xy^2 + 3y^3), the GCF is y^2. This gives us: x^2(4x + 3y) + y^2(4x + 3y).
03

Factor Out the Common Binomial

Now, observe that (4x + 3y) is a common binomial factor in both terms. Factoring out (4x + 3y) from the expression gives us: (4x + 3y)(x^2 + y^2).

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Key Concepts

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

polynomials
A polynomial is an expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. Polynomials can have multiple terms. For example, the given expression,
\(4x^3 + 3x^2y + 4xy^2 + 3y^3\), is a polynomial with four terms. Each term can include variables and constants raised to different powers.
Here are the key properties of polynomials:
  • Each term is a product of a constant coefficient and variables raised to varying powers.
  • The degree of a polynomial is the highest power of the variable in the polynomial. In our example, the highest power of any term is 3.
  • Polynomials can be added, subtracted, and multiplied.
greatest common factor
The Greatest Common Factor (GCF) is the highest factor that two or more numbers (or terms) share. When factoring by grouping, finding the GCF within each grouping helps simplify the polynomial.
In our example, we break the polynomial into two groups and find the GCF for each group:
  • For \(4x^3 + 3x^2y\), the GCF is \(x^2\).
  • For \(4xy^2 + 3y^3\), the GCF is \(y^2\).
\tBy factoring out these GCFs from each group, we simplify each part:
\(x^2(4x + 3y) + y^2(4x + 3y)\). This step is crucial because it reveals a common binomial factor in both groups.
binomials
A binomial is a polynomial with exactly two terms. For example, \(4x + 3y\) is a binomial. In our factor by grouping problem, we end up with:
  • First factor: \(x^2\)
  • Second factor: \(y^2\)
  • Common binomial factor: \(4x + 3y\)
The process of factoring by grouping often results in a common binomial factor that can be further factored out.
This allows us to write the polynomial as a product of binomials and other polynomials.
In the final step of our example, the expression \(x^2(4x + 3y) + y^2(4x + 3y)\) becomes:
\((4x + 3y)(x^2 + y^2)\). \tFacilitating tasks like simplifying expressions and solving polynomial equations.

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Most popular questions from this chapter

If a trinomial has a negative coefficient for the squared term, as in \(-2 x^{2}+11 x-12,\) it is usually easier to factor by first factoring out the common factor \(-1 .\) $$ \begin{aligned} -2 x^{2}+11 x-12 \\ =&-1\left(2 x^{2}-11 x+12\right) \\ =&-1(2 x-3)(x-4) \end{aligned} $$ Use this method to factor each trinomial. See Example 7(b). $$ $$ -2 a^{2}-5 a b-2 b^{2} $$

The middle term of each trinomial has been rewritten. Now factor by grouping. $$ \begin{aligned} 6 x^{2}+13 x+6 \\ =6 x^{2}+9 x+4 x+6 \end{aligned} $$

Factor by grouping. \(6-3 x-2 y+x y\)

Factor completely. If the polynomial cannot be factored, write prime. $$ -32+14 x+x^{2} $$

Factor each trinomial completely. $$ 4 z^{2}-12 z w+9 w^{2} $$
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