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Chapter 4: Problem 72

Simplify. $$7\left(x^{2}+3 x+1\right)+9\left(x^{2}+3 x+1\right)-5\left(x^{2}+3 x+1\right)$$

Short Answer

Expert verified
The simplified expression is \(11x^{2}+33x+11\).

Step by step solution

01

Applying Distributive Property

First, the distributive property allows for the operations inside the parentheses to be carried out with the constants outside. \n So, \(7(x^{2}+3 x+1)+9(x^{2}+3 x+1)-5(x^{2}+3 x+1)\) will result into \(7x^{2}+21x+7+9x^{2}+27x+9-5x^{2}-15x-5\).
02

Combining Like Terms

Next, identify and combine like terms. Like terms are terms with the same variable and exponent. So, \(7x^{2}+9x^{2}-5x^{2}\) simplifies to \(11x^{2}\), \(21x+27x-15x\) simplifies to \(33x\), and \(7+9-5\) simplifies to \(11\).
03

Final Expression

So, the simplified expression will result into \(11x^{2}+33x+11\).

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that helps in simplifying expressions. It allows you to multiply a single term by each term inside a set of parentheses. This is crucial when dealing with expressions that involve sums or differences. For instance, in the expression \(7(x^2+3x+1)\), the distributive property lets us express it as \(7x^2 + 21x + 7\).
  • Start by multiplying the term outside the parentheses by each term within the parentheses.
  • Keep track of the signs; multiplying by a negative reverses the signs of the terms.
  • Calculate each multiplication accurately to avoid errors later in the process.
Applying the distributive property simplifies complex expressions and prepares them for the next step, which is combining like terms.
Combining Like Terms
After distributing, the next step in simplifying the expression is combining like terms. Like terms have the same variable raised to the same power. It helps in reducing the expression to its simplest form. In our example, after distribution, the expression becomes \(7x^2 + 21x + 7 + 9x^2 + 27x + 9 - 5x^2 - 15x - 5\).
  • First, identify terms that have identical variables and exponents.
  • For instance, \(7x^2, 9x^2,\) and \(-5x^2\) are like terms because they all contain \(x^2\).
  • Similarly, the terms \(21x, 27x,\) and \(-15x\) can be combined.
  • Finally, the constant terms \(7, 9,\) and \(-5\) are also combined.
Combine these like terms to get a simplified form: \(11x^2 + 33x + 11\). This process enables further simplification or solving the equation if needed.
Polynomials
Polynomials are algebraic expressions made up of terms that are sums or differences of variables raised to a power, multiplied by coefficients. In the exercise, the expression \((x^2 + 3x + 1)\) is a polynomial.
  • A polynomial can consist of several terms. Each term is made up of a coefficient and a variable part.
  • The degree of a polynomial is determined by the highest power of the variable present.
  • In the expression \(x^2 + 3x + 1\), the highest degree is 2, indicating it's a quadratic polynomial.
Polynomials can be added, subtracted, multiplied, and factored, making them a versatile tool in algebra. When working with polynomials, understanding their structure helps in applying rules like the distributive property and combining like terms efficiently.

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Most popular questions from this chapter

Consider the following information: If a house was purchased for 105,000 dollar and is expected to appreciate 900 dollar per year, its value \(y\) after \(x\) years is given by the formula \(y=900 x+105,000 .\) A second house was purchased for 8120,000 and was expected to appreciate 1,000 dollar per year. Find a polynomial equation that will give the value of the house in \(x\) years.

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