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Chapter 6: Problem 194

Multiply. $$6\left(y^{2}+8 y+16\right)$$

Short Answer

Expert verified
6y^2 + 48y + 96

Step by step solution

01

Identify the Expression Inside the Parentheses

The given mathematical expression is: 6(y^2 + 8y + 16)
02

Distribute the Constant

Multiply each term inside the parentheses by the constant 6. 6 * y^2 + 6 * 8y + 6 * 16
03

Simplify Each Term

Perform the multiplication for each term: 6y^2 + 48y + 96

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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 helpful way of understanding how to multiply a single term with a polynomial.
It states that you need to multiply each term inside the parentheses by the term outside of them.
This applies not just to numbers but also to algebraic expressions.
For our exercise, we have the expression: 6(y^2 + 8y + 16). The distributive property indicates that each term inside the parenthesis鈥擻(y^2\), 8y, and 16鈥攕hould be separately multiplied by 6.
  • First term: 6 * y^2
  • Second term: 6 * 8y
  • Third term: 6 * 16
This makes it easier to manage complex expressions and carry multiplication across different terms.
Multiplying Polynomials
Multiplying polynomials involves distributing, similar to our previous concept.
When you multiply a polynomial by a monomial, we distribute the monomial over each term of the polynomial. For example, in our expression, we multiply 6 with each term inside:
6 * y^2, 6 * 8y, and 6 * 16.
Breaking it down:
  • First term multiplication: 6 * y^2 = 6y^2
  • Second term multiplication: 6 * 8y = 48y
  • Third term multiplication: 6 * 16 = 96
By performing these operations step-by-step, we simplify our work making it more manageable.
Simplifying Algebraic Expressions
Simplifying algebraic expressions means combining like terms and ensuring that the expression is in its simplest form.
After distributing the terms in our exercise, we got 6y^2 + 48y + 96. Here, each term is distinct, represented as powers of y and a constant.
  • There are no like terms (terms with the same variable raised to the same power) to combine.
  • Therefore, the expression is already in its simplest form once the distribution is completed and each term is multiplied.
Simplifying algebraic expressions helps clarify and streamline problem-solving processes and ensures the final result is neat and concise.

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Most popular questions from this chapter

Divide each polynomial by the binomial. $$\left(7 q^{3}-5 q-2\right) \div(q-1)$$

Divide each polynomial by the monomial. $$\frac{42 x^{2}-14 x}{7}$$

Divide each polynomial by the monomial. $$\frac{10 x^{2}+5 x-4}{-5 x}$$

Mixed Practice (a) \(24 a^{5}+2 a^{5}\) (b) \(24 a^{5}-2 a^{5}\) (c) \(24 a^{5} \cdot 2 a^{5}\) (d) \(24 a^{5} \div 2 a^{5}\)

Divide each polynomial by the monomial. $$\frac{20 b^{2}-12 b}{-4}$$
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