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Chapter 13: Problem 21

Given that $$ \frac{p(x)}{x-4}=x^{2}+7 x+10 $$ find $$ \frac{p(x)}{x^{2}+x-20} $$

Short Answer

Expert verified
Question: Find the value of \(\frac{p(x)}{x^{2}+x-20}\) if \(\frac{p(x)}{x-4}=x^{2}+7x+10\). Answer: \(\frac{p(x)}{x^{2}+x-20} = \frac{x^{3}+3x^{2}-18x-40}{x^{2}+x-20}\)

Step by step solution

01

Multiply the given polynomial expression

To find \(p(x)\), we multiply \((x-4)\) with \((x^{2}+7x+10)\): $$ p(x) = (x-4)(x^{2}+7x+10) $$
02

Expand the expression

Now let's expand the expression by distributing each term of \((x-4)\) to the terms of \((x^{2}+7x+10)\): $$ p(x) = x(x^2) + x(7x) + x(10) - 4(x^2) - 4(7x) - 4(10) $$
03

Simplify the expression

Simplify the product: $$ p(x) = x^{3} + 7x^{2} + 10x - 4x^{2} - 28x - 40 $$ Combine like terms: $$ p(x) = x^{3} + (7x^{2} - 4x^{2}) + (10x - 28x) - 40 $$ $$ p(x) = x^{3} + 3x^{2} - 18x - 40 $$
04

Insert p(x) into the desired expression

Now that we have found \(p(x)\), insert it into the expression we want to find: $$ \frac{p(x)}{x^{2}+x-20} = \frac{x^{3}+3x^{2}-18x-40}{x^{2}+x-20} $$ Since there is no straightforward simplification possible, our final answer is: $$ \frac{p(x)}{x^{2}+x-20} = \frac{x^{3}+3x^{2}-18x-40}{x^{2}+x-20} $$

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Key Concepts

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

Polynomial Multiplication
When we multiply polynomials, we essentially perform the operation of multiplying each term in one polynomial by each term in the other polynomial.
This process helps to expand expressions to discover the polynomial's simplified form. For instance, in the original exercise, we start with multiplying (x-4) and (x^2+7x+10) to get p(x).
Let's break down the multiplication:
  • The first term of the first polynomial, x, is multiplied by each term of the second polynomial resulting in \( x(x^2) + x(7x) + x(10) \)
  • The second term, -4, is also multiplied by each term in the second polynomial: \( -4(x^2) - 4(7x) - 4(10) \)
This shows how each contribution from the multiplication will impact the final expression.
Each multiplied term becomes part of the resulting expanded polynomial, which in our case is p(x). By multiplying out each term carefully, we ensure the polynomial expansion is accurate before moving onto the next step.
Polynomial Simplification
Simplifying a polynomial is about organizing and reducing an expression into simpler terms.
The goal is to make the expression as concise as possible.4 In our exercise, once p(x) is expanded, we start simplifying by combining terms.
This means that similar terms (terms with the same variable to the same power) are added or subtracted from each other.
For the polynomial:
  • The terms obtained by distribution were \( x^{3}, 7x^{2}, 10x, -4x^{2}, -28x, \) and \(-40 \)
  • We identify the like terms as \( x^{2} \) terms and \( x \) terms.
Combining these, we add 7x^2,鈭�4x^2 and we get 3x^2.
Similarly, we combine 10x and 鈭�28x to yield 鈭�18x. Doing these operations, we achieve a much simpler polynomial expression.
Combining Like Terms
Combining like terms is a crucial aspect of simplifying polynomials and involves identifying terms that can be added or subtracted based on their coefficients.
In practical terms, like terms are those that have the same variable and the same exponent.
This step reduces the complexity of a polynomial.
  • In the provided exercise, we dealt with the expression p(x) = x^{3} + 7x^{2} + 10x - 4x^{2} - 28x - 40
  • First, combine the \(x^2\) terms:\(7x^{2} - 4x^{2} = 3x^{2}\)
  • Next, we take care of the \(x\) terms:\(10x - 28x = -18x\)
Everything else remains unchanged since the term \( x^{3} \) and the constant term \(-40\) have no like terms to combine.
By ensuring that similar terms are combined, the polynomial is more manageable and clear to work with when further analyzing or utilizing the expression in calculations.

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Most popular questions from this chapter

Use the division algorithm to find the quotient \(q(x)\) and the remainder \(r(x)\) so that \(a(x)=\) \(q(x) b(x)+r(x)\). $$ \frac{5 x^{2}-23 x+16}{x-4} $$

Find the domain. $$ f(x)=\frac{x^{2}+x}{15 x} $$

Find the zeros. $$ f(x)=\frac{5 x^{2}-4 x-1}{10+3 x} $$

Use the Remainder Theorem to decide whether the first polynomial is a factor of the second. If so, what is the other factor? $$ x-3,3 x^{3}-13 x^{2}+10 x+9 $$

Solve for \(x\). $$ 7=\frac{5 x^{2}-2 x+23}{2 x^{2}+2} $$
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