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Chapter 7: Problem 115

Factor each expression. \(x^{2}-14 x-32\)

Short Answer

Expert verified
The factors are \ (x - 16)(x + 2)\.

Step by step solution

01

Identify the quadratic form

Recognize that the given expression is a quadratic equation in the form of \(ax^2 + bx + c\). Here, \(a = 1\), \(b = -14\), and \(c = -32\).
02

Find two numbers that multiply to c and add to b

Look for two numbers that multiply to \(-32\) (the constant term) and add to \(-14\) (the coefficient of \(x\)). These numbers are (-16) and 2\because \(-16 * 2 = -32\) and \(-16 + 2 = -14\).
03

Write the expression with these numbers

Rewrite the middle term (-14x) using the two numbers identified. The expression becomes \( x^2 - 16x + 2x - 32\).
04

Factor by grouping

Group the terms in pairs: \((x^2 - 16x) + (2x - 32)\). Factor out the common factor from each pair: \(x(x - 16) + 2(x - 16)\).

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Key Concepts

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

Quadratic Equations
Quadratic equations are algebraic equations of the form \(ax^2 + bx + c = 0\) where \(a, b,\) and \(c\) are constants and \(x\) represents an unknown variable.Quadratic equations have a characteristic 'U' shaped graph called a parabola. The solutions to a quadratic equation can be found using various methods including factoring, the quadratic formula, or completing the square.
Factoring
Factoring is the process of breaking down an expression into simpler expressions, or 'factors', that when multiplied together give the original expression.For the quadratic expression \(x^2 - 14x - 32\), we identified two numbers that multiply to the constant term \(-32\) and add up to the coefficient of \(x\), which is \(-14\). These numbers are \(-16\) and \(2\). Then, the expression was re-written using these numbers, and grouped in pairs to factor them out.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. Examples include \(3x + 4\) or \(x^2 - 5x + 6\). When working with algebraic expressions, we use algebraic rules and techniques, such as expanding, factoring, and simplifying, to manipulate them and solve for unknowns.
Polynomial Factorization
Polynomial factorization is a technique used in algebra to break down a polynomial into simpler 'factor' polynomials that, when multiplied together, form the original polynomial. In this case, for the polynomial \(x^2 - 14x - 32\), after identifying the numbers that multiply to \(-32\) and add to \(-14\), we re-wrote the expression and factored by grouping. The final factored form is \( (x - 16)(x + 2) \). This makes solving for \(x\) easier, as we can set each factor to zero and solve the resulting linear equations.

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