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Chapter 4: Problem 14

For the following problems, find the products. $$ (y+1)^{2} $$

Short Answer

Expert verified
Answer: The product of the given expression (y+1)^{2} is \(y^{2} + 2y + 1\).

Step by step solution

01

Identify the binomial

The given expression is (y+1)^{2}, which is a square of the binomial (y+1).
02

Square the first term

In the binomial (y+1), the first term is 'y'. Square of the first term will be y^{2}.
03

Square the second term

In the binomial (y+1), the second term is '1'. Square of the second term will be 1^{2}, which equals 1.
04

Find twice the product of the two terms

Multiply the two terms y and 1, i.e., y*1 = y, and find twice the product, i.e., 2(y*1) = 2y.
05

Add all the terms obtained

Add all the terms obtained in the previous steps, i.e, y^{2}, 1, and 2y. The sum will be: $$ y^2 + 1 + 2y $$ So, the product of the given expression, which is the square of the binomial (y+1), is \(y^{2} + 2y + 1\).

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Key Concepts

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

Understanding Binomial Expansion
Binomial expansion is a fundamental concept when working with algebraic expressions. It allows us to multiply expressions that involve two terms, known as binomials. The binomial theorem gives us a systematic method to expand expressions of the form \((a + b)^n\).

To comprehend binomial expansion, consider these steps:
  • Identify the binomial: A typical example is \((y + 1)^2\).
  • Apply the formula: The binomial theorem suggests an expansion method where powers are distributed over terms.
  • Add coefficients: These are derived using Pascal鈥檚 triangle or binomial coefficients.
This concept is useful for finding powers without directly multiplying, especially when the exponent is larger.
Exploring Polynomials
Polynomials are expressions that consist of variables raised to whole number powers, combined using addition, subtraction, and multiplication. Each separate expression within a polynomial is called a 'term.'

A polynomial can look like: \(y^2 + 2y + 1\). This particular expression is a quadratic polynomial since the highest power of the variable 'y' is 2.

Key features of polynomials include:
  • Terms: These are composed of a coefficient and variables raised to a power.
  • Degree: This is the highest power present in the expression, like '2' in our quadratic example.
  • Standard form: Arranging terms in descending order according to their degree, for easier comparison and evaluation.
Polynomials form the basis for various algebraic operations and are fundamental in calculus and other advanced topics.
The Basics of Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number being multiplied, and the exponent indicates how many times the base is used as a factor.

In the expression \((y + 1)^2\),
  • The base is \((y + 1)\), meaning this whole term gets multiplied by itself.
  • The exponent is '2', indicating the multiplication happens twice.
Main principles of exponentiation include:
  • Power of a product: \((ab)^n = a^n b^n\), meaning each factor is raised to the power.
  • Power of a power: \((a^m)^n = a^{mn}\), meaning powers multiply when a power is raised to another power.
Understanding these basics is essential for manipulating and simplifying expressions in algebra.

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Most popular questions from this chapter

Classify the polynomial given below as a monomial, binomial, trinomial, or none of these. Specify the degree of the polynomial and write the numerical coefficient of each term. \(3 x^{3} y+4 x y^{4}+8 x^{2} y^{2} z^{0} w, \quad z \neq 0\)

For the following problems, simplify each of the algebraic expressions. $$ 4 a-a(a+5) $$

For the following problems, perform the multiplications and combine any like terms. $$ y^{3}(y-3)(y-2) $$

For the following problems, simplify each of the algebraic expressions. $$ 2(x-6)+5 $$

For the following problems, answer the question of how many. $$ c^{3} \text { 's in } 2 a^{2} b c^{3} ? $$
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