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Chapter 2: Problem 37

Multiply out and simplify as completely as possible. $$3\left(x^{2}-4 x+7\right)$$

Short Answer

Expert verified
3x^2 - 12x + 21

Step by step solution

01

- Distribute 3 to each term inside the parentheses

To solve this exercise, multiply 3 by each term inside the parentheses: \(3 \cdot x^2\), \(3 \cdot (-4x)\), and \(3 \cdot 7\).
02

- Perform the multiplication

Calculate each multiplication individually: \(3 \cdot x^2 = 3x^2\), \(3 \cdot (-4x) = -12x\), and \(3 \cdot 7 = 21\).
03

- Combine the results

Now combine all the resulting terms: \(3x^2 - 12x + 21\).

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Key Concepts

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

algebraic multiplication
Algebraic multiplication is a fundamental concept in algebra. It involves multiplying numbers, variables, or a combination of both. When we talk about algebraic multiplication with polynomials, it involves multiplying each term in a polynomial by another number or polynomial.

In this exercise, we have the expression $$3(x^2-4x+7)$$. Multiplying a number through a polynomial is a typical example of algebraic multiplication.

To multiply, we distribute the number across each term in the polynomial. For example, with $$3(x^2-4x+7)$$, we multiply 3 by each term within the parentheses.
polynomials
A polynomial is an expression made up of variables, coefficients, and exponents. Typically, it features multiple terms separated by plus or minus signs.

In the exercise, we have the polynomial inside the parentheses: $$x^2-4x+7$$.

This polynomial has three terms:
  • The first term is $$x^2$$, where the coefficient is 1 and the exponent is 2.
  • The second term is $$-4x$$, with a coefficient of -4 and an exponent of 1.
  • The third term is 7, which is a constant term with a coefficient of 7 and no variable.
Understanding the structure of a polynomial helps us apply algebraic operations correctly, like multiplication in the initial question.
simplifying expressions
Simplifying an expression means to make it as concise as possible. This usually involves combining like terms and performing any basic arithmetic operations.

In the context of the original exercise, starting from $$3(x^2-4x+7)$$, we simplify by distributing the 3 across each term.

Step-by-step, this looks like:
  • Multiply 3 by $$x^2$$ to get $$3x^2$$
  • Multiply 3 by $$-4x$$ to get $$-12x$$
  • Multiply 3 by 7 to get 21


After performing these individual multiplications, we combine the results. The simplified expression is $$3x^2 - 12x + 21$$. This final expression is the simplest form of the given problem.

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