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Chapter 7: Problem 8

$$ x^{4}+14 x^{3}+71 x^{2}+154 x+120 . $$

Short Answer

Expert verified
The factorized form of the given polynomial \(x^{4}+14 x^{3}+71 x^{2}+154 x+120\) is \((x+2)(x+3)(x+4)(x+5)\)

Step by step solution

01

Identify the coefficients

The polynomial is \(x^{4}+14 x^{3}+71 x^{2}+154 x+120\) and so has the coefficients a = 1, b = 14, c =71, d = 154, and e =120.
02

Guess the factors

Since the constant term of the polynomial (120) is positive, possible roots might be its factors. After trying few factors, we find that -2, -3, -4, and -5 are the roots of the polynomial.
03

Confirmation and Factoring

Confirm the guessed roots by substitution in original polynomial. If the result is 0, then it is a root; so we have \(f(-2) = 0\), \(f(-3) = 0\), \(f(-4) = 0\), and \(f(-5) = 0\). This confirms that these are the roots. Therefore, the factors of the polynomial are \((x+2)\), \((x+3)\), \((x+4)\), and \((x+5)\). Hence, the given polynomial can be factored as \((x+2)(x+3)(x+4)(x+5)\).

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Key Concepts

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

Polynomial Roots
Understanding the roots of a polynomial is essential to fully grasp polynomial factoring. Roots, often referred to as zeroes, are the values of x that make a polynomial equal to zero. In other words, if you were to plug a root into the polynomial, the output would be zero.

For instance, in the exercise provided, the polynomial \( x^{4}+14 x^{3}+71 x^{2}+154 x+120 \) has roots at -2, -3, -4, and -5. This means that if you replace 'x' in the polynomial with any of these numbers, the result is zero. It's a bit like a magic trick; these special numbers cause the entire polynomial to disappear!
Factoring Polynomials
Factoring polynomials is a bit like breaking down a complex machine into its simpler parts. It involves rewriting the polynomial as a product of its factors, much as you might write \( 12 \)= \( 2 \times 3 \times 2 \) when dealing with numbers.

For polynomials, factors are usually in the form of binomials or trinomials. The example exercise shows that the polynomial \( x^{4}+14 x^{3}+71 x^{2}+154 x+120 \) can be expressed as the product of the binomials \( (x+2)(x+3)(x+4)(x+5) \). Each of these binomials contains a root of the polynomial, and when multiplied together, they combine to form the original polynomial.
Coefficient Identification
The first step in solving polynomials is often identifying the coefficients, which are the numbers in front of the variables. In our exercise, the coefficients are the numbers 1 (implied in front of \( x^{4} \) since any number times 1 is itself), 14, 71, 154, and 120. These coefficients hold the secret to the structure of the polynomial and unlock the path to finding its roots.

By analyzing the signs and values of these coefficients, we can predict possible roots since they are often factors of the constant term; in this case, 120.
Root Confirmation
Once you've made an educated guess about the roots of the polynomial, confirmation is a crucial step. This is done by substituting the guessed roots back into the original equation to see if they indeed produce zero.

In our example, we confirm our roots -2, -3, -4, and -5 by substituting them back into the original polynomial, \( x^{4}+14 x^{3}+71 x^{2}+154 x+120 \), to ensure they result in zero. Successfully confirming roots brings a sense of accomplishment, like finding the last piece of a puzzle. Without this step, we cannot be certain that we have factored the polynomial correctly.

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Most popular questions from this chapter

Find the nature of roots of the polynomial \(P(x)=3 x^{4}+12 x^{2}+5 x-4\).

$$ \frac{1}{x^{2}-3 x+3}+\frac{2}{x^{2}-3 x+4}=\frac{6}{x^{2}-3 x+5} $$

$$ \left(x^{2}-4\right) \sqrt{x+1}=0 $$

$$ (x-4)(x-5)(x-6)(x-7)=1680 $$

$$ 4 \cdot 3^{x+2}+5 \cdot 3^{x}-7 \cdot 3^{x+1}=40 $$
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