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Chapter 9: Problem 105

Expand. $$(y-3)^{2}$$

Short Answer

Expert verified
The expansion of \( (y-3)^{2} = y^{2} - 6y + 9 \)

Step by step solution

01

Write out the binomial theorem

The expansion of \( (a-b)^{2} \) according to the binomial theorem is \( a^{2} - 2ab + b^{2} \). In this case, \( a=y \) and \( b=3 \).
02

Apply the binomial theorem

Follow the binomial theorem we can substitute \( y \) for \( a \) and \( 3 \) for \( b \) in the formula, we get \( y^{2} - 2*y*3 + 3^{2}\).
03

Simplify the expression

We simplify the expression obtained in the previous step to get \( y^{2} - 6y + 9\).

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Key Concepts

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

Quadratic Expression
A quadratic expression is a polynomial expression of degree two. This means the highest exponent of the variable is 2. It generally has the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In the expression \((y-3)^2\), after expansion, we get \( y^2 - 6y + 9 \), which is a quadratic expression because the highest power of \( y \) is 2.

Quadratics can appear in different forms:
  • Standard form: \( ax^2 + bx + c \)
  • Vertex form: \( a(x-h)^2 + k \)
  • Factored form: \( a(x-p)(x-q) \)
Understanding these forms helps in solving quadratic equations efficiently. Often, binomial expressions like \((y-3)^2\) give rise to quadratic expressions when expanded. This is particularly useful in equations and functions seen in algebra.
Polynomial Expansion
Polynomial expansion is the process of multiplying out expressions inside parentheses to rewrite them as a standard polynomial. This tactic is often used for simplifying expressions to make calculations easier.

Take the expression \((y-3)^2\). When we expand this using the binomial theorem, we multiply each component by every other component as per the rule \((a-b)^2 = a^2 - 2ab + b^2\). Thus, \((y-3)^2\) expands to \( y^2 - 6y + 9 \).

Understanding expansion involves recognizing patterns and systematically applying algebraic rules. It allows you to transform complex-looking expressions into simpler, more recognizable polynomial forms.
Algebraic Identities
Algebraic identities are mathematical expressions that are universally true for all values of the involved variables. They are crucial tools in algebra for simplifying expressions and solving equations quickly. One of the key identities used in binomial expansion is \((a-b)^2 = a^2 - 2ab + b^2\).

This identity allows us to expand expressions such as \((y-3)^2\) without distributing each term individually. Knowing and recognizing these identities simplifies many algebraic computations and helps in expanding polynomials faster.

Common algebraic identities include:
  • \((a+b)^2 = a^2 + 2ab + b^2\)
  • \((a-b)^2 = a^2 - 2ab + b^2\)
  • \((a+b)(a-b) = a^2 - b^2\)
Mastering these identities empowers students to handle polynomial expansions with greater ease and confidence.

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Most popular questions from this chapter

$$(x+1)\left(-x^{6}+x^{5}-x^{4}+x^{3}-x^{2}+x-1\right)$$

Biologists and physicists are working together to measure the mass of a virus. Currently, a virus with a mass of 0.00000000000000000039 g can be measured. Write this number in scientific notation.

Expand. $$(x+1)^{2}$$

Multiply. $$(11 x+2 y)(3 x+7 y)$$

Multiply. $$(5 a-12)(3 a-7)$$
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