Problem 85 Multiply. $$ (4 a+1)^{2} $... [FREE SOLUTION]

Chapter 6: Problem 85

Multiply. $$ (4 a+1)^{2} $$

Short Answer

Expert verified

The expanded expression is \ 16a^2 + 8a + 1 \.

Step by step solution

- Identify the formula

Recognize that the given expression is in the form of a binomial squared $ (a+b)^2 $, where $ a = 4a $ and $ b = 1 $.

- Apply the binomial square formula

The binomial square formula is \[ (a+b)^2 = a^2 + 2ab + b^2 \]. Substitute the identified values into the formula: $(4a+1)^2 = (4a)^2 + 2(4a)(1) + 1^2$.

- Calculate each term

Calculate $ (4a)^2 $, which is \ 16a^2 \; 2(4a)(1), which is \ 8a \; and \ 1^2 \, which is \ 1 \.

- Combine the terms

Combine the calculated terms to get the final expanded expression: \[ 16a^2 + 8a + 1 \].

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Key Concepts

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

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators like addition and multiplication. They are used to represent quantities in algebra. For example, in the expression $4a + 1$, $4a$ and $1$ are terms.

Terms: Parts of the expression separated by $+$ or $-$ signs.
Constants: Numbers on their own, e.g. $1$.
Variables: Symbols representing numbers, e.g. \ 'a' \ in the term $4a$.

Understanding algebraic expressions helps in manipulating and solving them efficiently. They are foundational for more advanced topics like polynomial multiplication and the binomial theorem.

Polynomial Multiplication

Polynomial multiplication involves expanding expressions where variables are raised to powers and multiplied by coefficients. It includes operations like multiplying binomials and applying specific formulas.Here, we鈥檙e multiplying $4a + 1$ by itself.

Distributive Property: This property helps in expanding expressions by distributing each term of one polynomial to every term of another.
Foil Method: Specifically for binomials, it stands for First, Outer, Inner, Last terms multiplication.

When multiplying $(4a + 1)^2$, we use the distributive property or the binomial theorem to expand it.

Binomial Theorem

The binomial theorem provides a quick way to expand expressions raised to a power, such as $(4a + 1)^2$. It states $(a+b)^2 = a^2 + 2ab + b^2$. This formula is crucial when dealing with binomials squared or to any power.

Identify Components: For $ (4a+1)^2 $, let \ 'a' = 4a \ and \ 'b' = 1 \.
Apply Formula: \ (4a)^2 + 2(4a)(1) + 1^2 \.
Calculate: Results in \ 16a^2 + 8a + 1 \.

By understanding this formula and how to apply it, expanding binomials like $4a + 1$ becomes straightforward.

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91影视

Multiply. $$ (4 a+1)^{2} $$

Short Answer

Step by step solution

- Identify the formula

- Apply the binomial square formula

- Calculate each term

- Combine the terms

Key Concepts

Algebraic Expressions

Polynomial Multiplication

Binomial Theorem

One App. One Place for Learning.

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