Problem 95 Beginning algebra students somet... [FREE SOLUTION]

Chapter 0: Problem 95

Beginning algebra students sometimes make the following error when squaring a binomial: $$(x+5)^{2}=x^{2}+25$$ (a) Substitute a value for $x$ to verify that this is an error. (b) What is the correct expansion for $(x+5)^{2} ?$

Short Answer

Expert verified

The error is shown when $(x+5)^2 = 49$ but $x^2+25 = 29$; the correct expansion is $x^2 + 10x + 25$.

Step by step solution

Substitute a Value into the Incorrect Expansion

Let's choose a value for $x$. For this example, we'll use $x = 2$. Substitute $2$ into the incorrect equation:\[ (2+5)^2 = 2^2 + 25 \]Calculate both sides to see if they are equal.

Calculate the Left-Hand Side (LHS)

Substitute $x = 2$ into $(x+5)^2$:\[ (2+5)^2 = 7^2 = 49 \]The left-hand side equals $49$.

Calculate the Right-Hand Side (RHS)

Substitute $x = 2$ into $x^2 + 25$:\[ 2^2 + 25 = 4 + 25 = 29 \]The right-hand side equals $29$.

Compare the LHS and RHS

Compare the two results from Steps 2 and 3. LHS: 49 and RHS: 29. Since $49 eq 29$, the expression $x^2 + 25$ is incorrect.

Find the Correct Expansion for \$ (x+5)^2 \$

Use the binomial formula $(a+b)^2 = a^2 + 2ab + b^2$ with $a = x$ and $b = 5$:\[ (x+5)^2 = x^2 + 2(x)(5) + 5^2 \]Simplify the expression:\[ (x+5)^2 = x^2 + 10x + 25 \]

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Key Concepts

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

Binomial Expansion

Binomial expansion is an essential concept in algebra that involves expanding expressions of the form $(a+b)^n$, where $a$ and $b$ are any numbers or algebraic expressions. This is particularly important when dealing with quadratic expressions, such as those derived from squaring binomials.
The binomial expansion is based on a formula, which can be quickly applied:

For $(a+b)^2$, use the identity: $(a+b)^2 = a^2 + 2ab + b^2$
It helps in recomposing the original expression into a long form, which showcases each component of the operation.

Understanding this helps eliminate common mistakes in algebraic manipulation, such as omitting the cross-product term $(2ab)$ while expanding.

Squaring Binomials

Squaring binomials is a vital operation in algebra that involves multiplying a binomial by itself. This is a scenario often encountered in polynomial equations and algebraic simplification.
The genuine challenge for many learners is remembering to apply the complete formula:

For $(x+5)^2$, instead of simply distributing the powers, one should ensure to add all necessary components in the product:
\[ (x+5)^2 = x^2 + 2 \times x \times 5 + 5^2 = x^2 + 10x + 25 \]
The additional term 2ab (in this case, 10x) is often missed, leading to errors like in the original exercise.

Familiarity and practice with this process improve accuracy and confidence in handling complex algebraic expressions.

Polynomials

Polynomials are algebraic expressions made up of terms involving variables raised to different powers, typically summed together. They form the basis of many equations and problems in algebra.
A polynomial like $x^2 + 10x + 25$ contains various terms and helps in expressing the binomial square fully expanded or in standard form.
Key characteristics of polynomials include:

Each term consists of a coefficient, a variable, and a non-negative integer exponent.
They can have one or multiple terms, referred to as monomials, binomials, trinomials, etc., depending on the number of terms.
Operations like addition, subtraction, multiplication, and division can apply, keeping results within the polynomial structures.

The expanded form of a squared binomial is often expressed as a polynomial, helping clearly define the resultant expression.

Algebraic Error Correction

Algebraic errors, especially when expanding binomials or working with polynomials, are frequently encountered by learners. Correcting these mistakes involves identifying the missing or incorrect components within an operation.
In the original exercise, the mistake occurred when squaring a binomial and forgetting the cross-term $2ab$:

The incorrect expansion was $x^2 + 25$, which lacked the middle term $10x$.
Verification showed a discrepancy in calculated values, confirming an error in the initial assumption.

To prevent such errors:

Always apply the full expansion formula $(a+b)^2 = a^2 + 2ab + b^2$ when working with squared binomials.
Double-check each step to ensure all calculations align with the intended expressions.

Developing precision in algebra fosters both skill and confidence in mathematical operations.

91影视

Beginning algebra students sometimes make the following error when squaring a binomial: $$(x+5)^{2}=x^{2}+25$$ (a) Substitute a value for \(x\) to verify that this is an error. (b) What is the correct expansion for \((x+5)^{2} ?\)

Short Answer

Step by step solution

Substitute a Value into the Incorrect Expansion

Calculate the Left-Hand Side (LHS)

Calculate the Right-Hand Side (RHS)

Compare the LHS and RHS

Find the Correct Expansion for \\( (x+5)^2 \\)

Key Concepts

Binomial Expansion

Squaring Binomials

Polynomials

Algebraic Error Correction

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