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Chapter 6: Problem 40

Write in factored form by factoring out the greatest common factor. \(25 k^{4}+15 k^{2}\)

Short Answer

Expert verified
The factored form is \(5k^2(5k^2 + 3)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, determine the greatest common factor of the coefficients and the variables in the terms. The coefficients are 25 and 15, and their GCF is 5. For the variables, we have \(k^4\) and \(k^2\). The GCF of \(k^4\) and \(k^2\) is \(k^2\). Therefore, the GCF of the entire expression is \(5k^2\).
02

Factor out the GCF

Next, factor \(5k^2\) out from each term in the expression \(25k^4 + 15k^2\): \[25k^4 + 15k^2 = 5k^2(5k^2) + 5k^2(3)\]. Take \(5k^2\) common: \[25k^4 + 15k^2 = 5k^2(5k^2 + 3)\].

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Key Concepts

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

Greatest Common Factor
In mathematics, the greatest common factor (GCF) is the highest number that divides exactly into two or more numbers. When dealing with polynomials, the GCF also applies to variable parts. To find the GCF of polynomial terms, look for both numerical and variable components:
  • Identify the GCF of the coefficients: For 25 and 15, it is 5.
  • Identify the GCF of the variables: For \(k^4\) and \(k^2\), it is \(k^2\).
In our exercise, the GCF of the polynomial \(25 k^{4}+15 k^{2}\) is \(5k^2\). This makes the first step in factoring the polynomial finding this common factor.
Factored Form
Factored form of a polynomial expression involves expressing it as a product of its factors. This simplifies the process of solving equations and understanding their roots. To factor out the GCF:
  • Write the original polynomial.
  • Divide each term by the GCF.
  • Express the original polynomial as a product of the GCF and the simplified polynomial.
For \(25 k^{4}+15 k^{2}\), we factor out \(5k^2\) to get: \[25k^4 + 15k^2 = 5k^2(5k^2 + 3)\]. Now, \(5k^2 (5k^2 + 3)\) is the polynomial in its factored form.
Polynomial Expressions
Polynomial expressions are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A basic polynomial expression example is \(25k^4 + 15k^2\). Polynomials come in various degrees based on the highest power of the variable.
  • To simplify and solve polynomials, factoring can be very useful.
  • It involves breaking down complex expressions into products of simpler ones.
In our exercise with \(25k^4 + 15k^2\), recognizing the structure and degree helps in understanding how to manipulate and simplify the expression by factoring. Factoring reduces a polynomial to the simplest form, making it easier to handle in further algebraic operations. This skill is essential for advancing in algebra and beyond.

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Most popular questions from this chapter

Factor completely. $$ t^{2}-t z-6 z^{2} $$

Galileo 's formula describing the motion of freely falling objects is $$ d=16 t^{2} $$ The distance d in feet an object falls depends on the time \(t\) elapsed, in seconds. (This is an example of an important mathematical concept, the function.) When you substituted 256 for \(d\) and solved the formula for \(t\) in Exercise \(79,\) you should have found two solutions: 4 and \(-4 .\) Why doesn't \(-4\) make sense as an answer?

Find the greatest common factor for each list of terms. \(16 y, 24\)

Brain Busters Factor each polynomial. ( Hint: As the first step, factor out the greatest common factor.) $$ 25 q^{2}(m+1)^{3}-5 q(m+1)^{3}-2(m+1)^{3} $$

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} $$
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