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Chapter 8: Problem 7

Factor each of the following expressions as completely as possible. If an expression is not factorable, say so. $$x^{2}+3 x-2$$

Short Answer

Expert verified
\[(x - 1)(x + 4)\]

Step by step solution

01

Identify coefficients

Identify the coefficients of the quadratic expression in the standard form quadratic expression: \[x^{2} + 3x - 2\] This expression is in the form \[ax^{2} + bx + c\]. Here, \[a = 1,\] \[b = 3,\] \[c = -2\].
02

Find the product of ac

Multiply the coefficient of \(x^2\) (which is a) and the constant term (which is c): \[a \times c = 1 \times -2 = -2\].
03

Find factors of ac that sum to b

Look for two numbers that multiply to \[-2\] and add up to \[b = 3\]. The numbers are \[4\] and \[-1\] because \[4 \times -1 = -2\] and \[4 + (-1) = 3\].
04

Rewrite the middle term

Split the middle term \(3x\) into \[4x - x\] using the factors found: \[x^2 + 4x - x - 2.\]
05

Factor by grouping

Group terms to factor each part separately: \[(x^2 + 4x) - (x + 2).\]Factor out the greatest common factor from each group: \[x(x + 4) - 1(x + 4).\]
06

Factor out the common binomial factor

Factor out \(x + 4\): \[(x - 1)(x + 4)\]
07

Final Factored Form

The final factored form of the expression \[x^2 + 3x - 2\] is: \[(x - 1)(x + 4).\]

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Key Concepts

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

quadratic expression
A quadratic expression is a polynomial of degree 2.
This means the highest power of the variable (usually x) is 2.
Quadratic expressions have the form: \[ax^2 + bx + c\].
Here, a, b, and c are constants, and a is not zero.

In the example, \[x^2 + 3x - 2\], we identify:
  • a = 1
  • b = 3
  • c = -2
Recognizing a quadratic expression helps us apply specific techniques like factoring.
factoring by grouping
Factoring by grouping is a method used to factor some polynomials.
It involves rearranging terms into groups that can each be factored separately.

Here's how it works step by step for \[x^2 + 3x - 2\]:
  • Split the middle term using two numbers that multiply to a * c (-2) and add up to b (3).
    In our case, 4 and -1 work.
    So, rewrite the expression as\[x^2 + 4x - x - 2\].
  • Group the terms:\[(x^2 + 4x) + (-x - 2)\].
  • Factor each group:
    \[x(x + 4) - 1(x + 4)\].
  • Factor out the common binomial factor:
    \[x + 4\].
  • Combine to get: \[(x - 1)(x + 4)\].
Factoring by grouping simplifies finding the factors of a quadratic expression.
standard form of quadratic
The standard form of a quadratic equation is \[ax^2 + bx + c = 0\].
It's helpful because it sets a framework for solving quadratics using various methods like factoring, completing the square, or using the quadratic formula.

When we have a quadratic expression like \[x^2 + 3x - 2\]:
  • It's already in standard form \[ax^2 + bx + c\].
  • We identified the coefficients as:
    \[a = 1,\ b = 3,\ c = -2\].
By knowing the standard form, we can systematically approach solving or factoring the quadratic expression.

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Most popular questions from this chapter

Simplify the expression and express your final answer with positive exponents only. $$\left(x^{-3}\right)^{-4}$$

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