/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Add $$ \left(x^{2}-5 x+4\rig... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 10

Add $$ \left(x^{2}-5 x+4\right)+(8 x-9) $$

Short Answer

Expert verified
\(x^2 + 3x - 5\)

Step by step solution

01

Identify the terms

Identify and write down the terms from both expressions inside the parentheses. First expression: \(x^{2}-5x+4\). Second expression: \(8x-9\).
02

Combine like terms

Combine the terms with the same variable powers from both expressions. For this exercise, combine the coefficients of like terms. The expressions are rewritten as: \(x^2 - 5x + 4 + 8x - 9\).
03

Simplify the expression

Add or subtract the coefficients of like terms. Rearrange the terms: \(x^2+(-5x+8x)+(4-9)\). Simplify: \(x^2 + 3x - 5\).

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Key Concepts

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

combining like terms
When working with algebraic expressions, it's essential to understand 'like terms'. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(x^2 - 5x + 4 + 8x - 9\), 'like terms' are \( -5x \) and \( 8x \) because they both have the variable 'x' raised to the first power. Identifying these terms is the first step towards combining them.

To combine like terms:
  • Identify terms with the same variables and exponents.
  • Add or subtract their coefficients. For instance, -5x and 8x can be combined by adding their coefficients: -5 + 8 = 3.
By combining like terms, we reduce clutter and simplify the expression, making it easier to work with in later stages of solving an equation.
simplifying expressions
After combining like terms, the next essential step is simplifying the expression. Simplifying means reducing the expression to its most basic form.
For our example, after combining the like terms, we have \(x^2 + 3x - 5\). Here are the steps to simplify any algebraic expression:
  • Remove any parentheses by distributing any coefficients if necessary.
  • Combine like terms, as discussed in the previous section.
  • Arrange the terms in a standard form, usually descending powers of the variable(s).
Simplified expressions are essential as they are more manageable and easier to interpret, especially when solving equations or further manipulating the expression.
polynomials
Understanding polynomials and their structure is crucial for working in algebra. A polynomial is a sum of multiple terms where each term consists of a variable raised to a non-negative integer power and a coefficient. For instance, \(x^2 - 5x + 4 + 8x - 9\) is a polynomial.

Characteristics of polynomials include:
  • Each term is made up of a coefficient and a variable raised to an exponent.
  • The degree of a polynomial is determined by the term with the highest exponent. For example, in \(x^2 + 3x - 5\), the highest exponent is 2, so it鈥檚 a second-degree polynomial.
Polynomials can be classified based on their number of terms:
  • Monomial: single term (e.g., \(2x\)).
  • Binomial: two terms (e.g., \(-5x + 3\)).
  • Trinomial: three terms (e.g., \(x^2 + 3x - 5\)).
Recognizing these fundamental aspects of polynomials helps in simplifying, combining, and solving polynomial equations.

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Most popular questions from this chapter

Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D4 contains the formula $$ =2 * \mathrm{A} 4+3 * \mathrm{B} 4 $$ What is the value in \(\mathrm{D} 4\) if the value in \(\mathrm{A} 4\) is 5 and the value in \(\mathrm{B} 4\) is \(10^{2}?\)

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=3 x^{2}\\\ &g(x)=\frac{1}{x-9} \end{aligned} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=\frac{2}{x}\\\ &g(x)=x^{2}-4 \end{aligned} $$

Simplify. \(\left(3 x^{2}\right)^{3}+4 x^{2} \cdot 4 x^{4}-x^{4}(2 x)^{2}+\left((2 x)^{2}\right)^{3}-\) \(100 x^{2}\left(x^{2}\right)^{2}\)

If \(f(x)=c,\) where \(c\) is some positive constant, describe how the graphs of \(y=g(x)\) and \(y=(f+g)(x)\) will differ.
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