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Chapter 5: Problem 19

Factor out the GCF in each polynomial. $$ 2 x(z+7)+(z+7) $$

Short Answer

Expert verified
The factored form is \((z+7)(2x + 1)\).

Step by step solution

01

Identify the Common Factor

Look at each term in the expression \(2x(z+7) + (z+7)\) and identify the common factor between them. Notice that both terms contain \((z+7)\).
02

Factor Out the Common Factor

Since \((z+7)\) is common to both parts of the expression, factor it out. Rewrite the polynomial as \((z+7)(2x + 1)\).
03

Verify the Factoring

To ensure the factoring is correct, distribute \((z+7)\) back through \((2x + 1)\) to see if you retrieve the original polynomial: \((z+7)(2x) + (z+7)(1) = 2x(z+7) + (z+7)\). This confirms the factoring is accurate.

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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, often abbreviated as GCF, is an essential concept when it comes to factoring polynomials. The GCF is simply the largest factor that divides each term in a polynomial without a remainder. Identifying the GCF helps simplify expressions, making them easier to work with.

To determine the GCF of a polynomial, follow these steps:
  • List out the factors of each term in the polynomial.
  • Identify the common factors shared by each term.
  • Choose the largest of these common factors.
Using the GCF is crucial for efficient algebraic problem solving and plays a fundamental role in reducing polynomial expressions.
Polynomial Expressions
Polynomial expressions are mathematical phrases that can include variables, constants, and coefficients, connected by addition, subtraction, and multiplication. These expressions are a cornerstone of algebra and mathematics in general.

A typical polynomial is represented as a sum of terms, where each term is a product of a constant (coefficient) and a variable raised to a non-negative integer power. For example, in the polynomial expression \(2x^3 + 3x^2 + 4x + 5\), each part like \(2x^3\), \(3x^2\), \(4x\), and \(5\) is a term.
  • Degrees: The degree of a polynomial is determined by the highest power of its variables.
  • Coefficients: These are the numbers that multiply the variables or powers of variables.
Understanding polynomial expressions lays the groundwork for tackling more complex algebraic structures and operations.
Algebraic Factoring Methods
Algebraic factoring involves breaking down polynomials into simpler components that are multiplied together. Factoring is essential for solving polynomial equations and simplifying expressions.

There are several methods used to factor polynomials:
  • Greatest Common Factor (GCF): Extracting the GCF from each term, as seen in the problem where \((z+7)\) was factored out.
  • Factoring by Grouping: This method is useful for polynomials with four or more terms. It involves rearranging and grouping terms to reveal common factors.
  • Trinomials: Factoring trinomials involves looking for two numbers that multiply to give the product of the first and last coefficients and add to give the middle coefficient.
  • Special Products: Recognizing patterns such as difference of squares or perfect square trinomials can significantly simplify factored forms.
Mastering these factorization techniques is vital for success in algebra, as many higher-level mathematical concepts require a firm grasp of factoring.

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Most popular questions from this chapter

Factor each polynomial completely. See Examples 1 through 12. $$ (x-3)^{2}-2(x-3)-8 $$

Factor each polynomial completely. See Examples 1 through 12. $$ 12 x^{2}-17 x+6 $$

Recall that a graphing calculator may be used to check addition, subtraction, and multiplication of polynomials. In the same manner, a graphing calculator may be used to check factoring of polynomials in one variable. For example, to see that $$ 2 x^{3}-9 x^{2}-5 x=x(2 x+1)(x-5) $$ graph \(\mathrm{Y}_{1}=2 x^{3}-9 x^{2}-5 x\) and \(\mathrm{Y}_{2}=x(2 x+1)(x-5) .\) Then trace along both graphs to see that they coincide. Factor the following and use this method to check your results. $$ -6 x^{4}+10 x^{3}-4 x^{2} $$

Find the value of \(c\) that makes each trinomial a perfect square trinomial. $$ n^{2}-2 n+c $$

Suppose that an object is thrown upward with an initial velocity of 64 feet per second off the edge of a 960 -foot cliff. The height \(h(t)\) in feet of the object after \(t\) seconds is given by the function $$ h(t)=-16 t^{2}+64 t+960 $$ a. Find the height of the object at \(t=0\) seconds, \(t=3 \mathrm{sec}\) onds, \(t=6\) seconds, and \(t=9\) seconds. b. Explain why the height of the object increases and then decreases as time passes. c. Factor the polynomial \(-16 t^{2}+64 t+960\).
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