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Chapter 8: Problem 17

Factor each of the following as completely as possible. If the polynomial is not factorable, say so. $$2 y^{2}-y-6$$

Short Answer

Expert verified
(y - 2)(2y + 3)

Step by step solution

01

Identify the polynomial structure

The given polynomial is quadratic: \(2y^2 - y - 6\). It has the general form of a quadratic equation: \(ax^2 + bx + c\), where \(a = 2\), \(b = -1\), and \(c = -6\).
02

Multiply the leading coefficient by the constant term

Multiply the leading coefficient \(a\) and the constant term \(c\). In this case, multiply \(2\) and \(-6\): \(2 \times -6 = -12\).
03

Find two numbers that multiply to the product and sum to the middle coefficient

We need to find two numbers that multiply to \(-12\) and sum to \(-1\) (the middle coefficient). These numbers are \(-4\) and \(3\) since \(-4 \times 3 = -12\) and \(-4 + 3 = -1\).
04

Rewrite the middle term using the two numbers

Rewrite the polynomial by splitting the middle term \(-y\) using \(-4\) and \(3\): \(2y^2 - 4y + 3y - 6\).
05

Factor by grouping

Group the terms in pairs: \((2y^2 - 4y) + (3y - 6)\). Factor out the greatest common factor (GCF) from each pair. From the first group, factor out \(2y\): \(2y(y - 2)\). From the second group, factor out \(3\): \(3(y - 2)\).
06

Factor out the common binomial

The expression can now be written as \(2y(y - 2) + 3(y - 2)\). Factor out the common binomial factor \((y - 2)\): \((y - 2)(2y + 3)\).

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Key Concepts

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

Quadratic Equations
A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c \). Here, \(a \), \(b \), and \(c \) are constants, where \(a \) should never be zero as it would then not be quadratic. Quadratic equations can appear in many different forms, such as \(2y^2 - y - 6 \).

To solve quadratic equations, we typically use techniques like:
  • Factoring
  • Completing the square
  • Quadratic formula: \[ x = \frac {-b \pm \sqrt {b^2 - 4ac}}{2a} \]

In this problem, we're using factoring because it's usually the simplest method for simpler quadratics.
Polynomial Factorization
Polynomial factorization is the process of expressing a polynomial as a product of its factors.

These factors can be other polynomials or simpler expressions.
For example, for the quadratic polynomial \(2y^2 - y - 6 \), we see that it factors into \((y - 2)(2y + 3) \). Factoring helps in solving polynomial equations and understanding their roots more easily.

Here are some points to remember:
  • Identify the basic structure of the polynomial.
  • Look for common factors.
  • Use techniques like grouping and splitting terms to factorize fully.

Knowing the basic steps and practicing it repeatedly is key to mastering polynomial factorization.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (like +, -, *, and /). In the quadratic equation given in the exercise, \(2y^2 - y - 6 \), we have:

  • Variables: \(y \)
  • Coefficients: \(2 \), \(-1 \) (for \(-y \)), and \(-6 \)

Combining like terms and simplifying expressions are fundamental toward solving algebraic equations. Breaking down these expressions into manageable parts makes it easier to find solutions.

Remember:
  • Always follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication, and Division (left-to-right), Addition and Subtraction (left-to-right))
  • Simplify step by step to avoid mistakes.

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Most popular questions from this chapter

Factor each of the following as completely as possible. If the expression is not factorable, say so. Try factoring by grouping where it might help. $$x^{2}-x y-4 x+4 y$$

Divide each of the following. Use the long division process where necessary. $$\frac{3 x^{3}+14 x^{2}+2 x-4}{3 x+2}$$

Factor each of the following expressions as completely as possible. If an expression is not factorable, say so. $$4 x^{2}+40 x y+100 y^{2}$$

Divide each of the following. Use the long division process where necessary. $$\frac{2 x^{2}-3 x+7}{x-1}$$

Factor each of the following expressions as completely as possible. If an expression is not factorable, say so. $$6 x^{2}-30 x+36$$
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