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Chapter 5: Problem 26

Complete each factorization. $$z^{2}-3 z-10=(z+\quad)(z-\quad)$$

Short Answer

Expert verified
The complete factorization for the equation \(z^{2}-3 z-10\) is \( (z+2)(z-5)\).

Step by step solution

01

Identify the Coefficient of the quadratic equation

The first step is to identify the coefficients of the given equation. For the equation \(z^{2}-3z-10=0\), the coefficients are a=1 (coefficient of \(z^{2}\)), b=-3 (coefficient of z) and c=-10 (constant term).
02

Identify the values that will complete the binomials

Now, look for two numbers that multiplies to \(ac= (1) (-10)= -10\) and adds up to b=-3. The numbers are -5 and 2. These numbers will complete the binomials.
03

Write down the binomials

Write down the binomials using these numbers. Since one number is negative and the other one is positive, one binomial will be summed and the other will be subtracted. Therefore the factored form of the equation is: \(z^{2}-3 z-10=(z+2)(z-5)\).

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Key Concepts

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

Polynomial Factorization
Polynomial factorization is a method used in algebra to express a polynomial as a product of simpler polynomials, often called factors. Understanding how to factor polynomials is crucial because it simplifies equations and expressions, making them easier to work with.
  • Factors are polynomials that can be multiplied together to produce the original polynomial.
  • Factoring helps solve polynomial equations and simplifies algebraic expressions.
  • It can reveal more information about the polynomial, such as its zeros and behavior on a graph.
The goal of polynomial factorization is to break down complex polynomials into linear or quadratic factors as much as possible. For instance, a quadratic polynomial can often be factored into two binomials. By practicing factorization, students develop a deeper understanding of variable relationships and polynomial equations.
Quadratic Equations
Quadratic equations are polynomials where the highest degree of the variable, usually denoted as x or z, is two. One standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
  • The term \(ax^2\) is known as the quadratic term.
  • The term \(bx\) is the linear term.
  • The term \(c\) is the constant term.
Quadratic equations can be solved by various methods, such as:- Factoring- Completing the square- Using the quadratic formulaIn the example given, the equation \(z^2 - 3z - 10 = 0\) is a quadratic because the variable z is raised to the power of two. Solving by factoring involves expressing the quadratic equation in the form of two binomial expressions.
Binomial Expressions
Binomial expressions are a type of polynomial that contain exactly two terms. They take the general form \((ax + b)\) and \((cx + d)\). Binomials are significant because they are commonly used in the process of factoring quadratic equations.In our example, the quadratic equation \(z^2 - 3z - 10 = 0\) is factored into binomials as \((z + 2)\) and \((z - 5)\). Here is how these binomials are formed:
  • A binomial like \((z + 2)\) has two terms: \(z\) and the constant \(2\).
  • The other binomial \((z - 5)\) consists of \(z\) and the constant \(-5\).
Together, these binomials multiply to recreate the original quadratic, showing the relationship between the factors and the polynomial. Understanding binomials is vital for factoring quadratics, as it allows us to solve equations and simplify expressions efficiently.

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Most popular questions from this chapter

Factor. If the polynomial is prime, so indicate. $$3 b^{2}+3 a^{2}-a b$$

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$r^{2}-2 r s+4 s^{2}$$

Factor. If the polynomial is prime, so indicate. $$6 s^{5}-26 s^{4}-20 s^{3}$$

Factor. $$a^{2}+12 a b+36 b^{2}$$

Factor. $$16 x^{2}+8 x y+y^{2}-9$$
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