Problem 73 Factor completely. Identify any ... [FREE SOLUTION] (2024)

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Chapter 6: Problem 73

Factor completely. Identify any prime polynomials. $$ 25 w^{3}-10 w^{2}+w $$

Short Answer

Expert verified

The completely factored form is \(w(5w-1)^2\).

Step by step solution

02

Factor the Quadratic Expression

Factor the quadratic expression inside the parentheses: \(25w^2 - 10w + 1\). Attempt to factor by grouping or through the use of the quadratic formula. Identify two numbers that multiply to \(25 \times 1 = 25\) and add to \(-10\). These numbers are \(-5\) and \(-5\).\[25w^2 - 10w + 1 = (5w - 1)(5w - 1)\]

03

Combine the Factors

Write the expression as a product of all identified factors. Combine the GCF and the factored quadratic.\[25w^3 - 10w^2 + w = w(5w - 1)^2\]

04

Identify Any Prime Polynomials

Check if any further factorization is possible. The quadratic factors are already in their simplest form, meaning the factors are prime. Conclude the factorization process since no further factorization is possible.

Key Concepts

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

Greatest Common Factor

To begin factoring a polynomial, we first look for the greatest common factor (GCF). The GCF is the largest factor that divides all terms of the polynomial. In the given polynomial \(25w^3 - 10w^2 + w\), we identify the common factor in each term.

  • The terms are: \(25w^3\), \(-10w^2\), and \(w\).
  • Each term has a common factor of \(w\).

By factoring \(w\) out of each term, we simplify the expression:
\(25w^3 - 10w^2 + w = w(25w^2 - 10w + 1)\).
This step makes it easier to address more complex portions of the polynomial.

Quadratic Expression

After factoring out the GCF, we focus on the quadratic expression inside the parentheses: \(25w^2 - 10w + 1\). Quadratic expressions are polynomials of degree 2, typically in the form \(ax^2 + bx + c\).
Here:

  • \(a = 25\) (the coefficient of \(w^2\))
  • \(b = -10\) (the coefficient of \(w\))
  • \(c = 1\) (the constant term)

Factoring a quadratic expression typically involves finding two binomials whose product gives back the quadratic expression. We find factors that multiply to \(a \times c = 25 \times 1 = 25\) and add to \(b = -10\). The numbers \(-5\) and \(-5\) fit these conditions: \(25w^2 - 10w + 1 = (5w - 1)(5w - 1) = (5w - 1)^2\).

Prime Polynomial

A prime polynomial cannot be factored further using integer coefficients. In this problem, after we factor out GCF and rewrite the quadratic expression, we need to check if it’s a prime polynomial.
We did this by splitting \(25w^2 - 10w + 1\) into two binomials \((5w - 1)(5w - 1)\). Each binomial here is already in its simplest form, showcasing that the quadratic factors of the original polynomial are already prime.
No further factoring is possible, indicating that all the polynomials involved are prime in their respective form.

Factorization Steps

The steps needed to factorize any polynomial can simplify the process considerably:

  • Step 1: Identify and factor out the greatest common factor (GCF).
  • Step 2: Address any remaining quadratic expressions by factoring further.
  • Step 3: Combine all factors identified in each step.
  • Step 4: Check for prime polynomials to confirm if any further factorization is necessary.

Using these steps, we approach factorization methodically:

  • We started with \(25w^3-10w^2+w\).
  • Factored out \(w\) as GCF: \(25w^3 - 10w^2 + w = w(25w^2 - 10w + 1)\).
  • Factored the quadratic: \(25w^2 - 10w + 1 = (5w - 1)(5w - 1)\).
  • Combined all parts: \(w(5w - 1)^2\).

The result is an elegant and comprehensible factorization of the original polynomial.

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Problem 73 Factor completely. Identify any ... [FREE SOLUTION] (3)

Most popular questions from this chapter

Instead of using the zero product property, use the properties of equality tosolve \(4(3 x+5)=0\).An expression is \((x-4)(x-3)\). Evaluate this expression when \(x=4\).Factor completely. Identify any prime polynomials. $$ 7 x^{2}-63 y^{2} $$The width of a rectangle is \(6 \mathrm{ft}\) less than its length. Its area is\(112 \mathrm{ft}^{2}\).Solve. $$ d(7-d)=0 $$
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Problem 73 Factor completely. Identify any ... [FREE SOLUTION] (2024)

FAQs

What is the factor theorem solution? ›

According to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and 'a' is any real number, then, (x-a) is a factor of f(x), if f(a)=0. Also, we can say, if (x-a) is a factor of polynomial f(x), then f(a) = 0. This proves the converse of the theorem.

What does it mean when it says factor completely? ›

We say that a polynomial is factored completely when we can't factor it any more. Here are some suggestions that you should follow to make sure that you factor completely: Factor all common monomials first. Identify special products such as difference of squares or the square of a binomial.

What is an example of factoring? ›

Factor expressions, also known as factoring, mean rewriting the expression as the product of factors. For example, 3x + 12y can be factored into a simple expression of 3 (x + 4y). In this way, the calculations become easier. The terms 3 and (x + 4y) are known as factors.

What is the factoring formula? ›

Factoring formulas are used to write an algebraic expression as the product of two or more expressions. Some important factoring formulas are given as, (a + b)2 = a2 + 2ab + b. (a - b)2 = a2 - 2ab + b.

What is a factoring solution? ›

Factoring is a financial transaction and a type of debtor finance in which a business sells its accounts receivable (i.e., invoices) to a third party (called a factor) at a discount. A business will sometimes factor its receivable assets to meet its present and immediate cash needs.

What does it mean to factor a problem? ›

Definitions: Factoring a polynomial is expressing the polynomial as a product of two or more factors; it is somewhat the reverse process of multiplying. To factor polynomials, we generally make use of the following properties or identities; along with other more techniques. Distributive Property: ab+ac=a(b+c)

How do you find the factors of a problem? ›

The factors of 18 are 1, 2, 3, 6, 9, and 18. We can find the factors of a number by dividing the number by all possible divisors. To find all the factors of a number n using the division method, divide the number by all the natural numbers less than n. Identify the numbers that completely divide the given number.

How do you factor each polynomial completely? ›

Step 1: Group the first two terms together and then the last two terms together. Step 2: Factor out a GCF from each separate binomial. Step 3: Factor out the common binomial. Note that if we multiply our answer out, we do get the original polynomial.

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