Problem 5 A student incorrectly factored. ... [FREE SOLUTION]

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Chapter 5: Problem 5

A student incorrectly factored. $$ x^{2}+4 \text { as }(x+2)^{2}$$

Short Answer

Expert verified

The expression $x^2 + 4$ cannot be factored into real numbers because it's a sum of squares.

Step by step solution

Observe the Given Expression

Look at the expression to be factored: $x^2 + 4$

Review the Incorrect Factorization

Examine the incorrect factorization provided: $(x + 2)^2$

Expand the Incorrect Factorization

Expand $(x + 2)^2$ to see if it matches the original expression:\[(x + 2)^2 = x^2 + 4x + 4\]

Compare the Expanded Form with the Original Expression

Compare $x^2 + 4x + 4$ with the original expression $x^2 + 4$. Notice that the middle term $4x$ is not present in the original expression.

Conclude the Correct Factorization

Recognize that $x^2 + 4$ is a sum of squares. Recall that a sum of squares does not factor into real numbers. Therefore, the correct factorization does not exist in the set of real numbers.

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Key Concepts

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

Incorrect Factorization

In algebra, factorizing an expression means breaking it down into products of simpler expressions. However, sometimes we might incorrectly factorize an expression. Let's consider the expression given in the problem: $x^2 + 4$. The student incorrectly factored this expression as $(x + 2)^2$. It's important to check if this factorization is correct. To do that, we expand the binomial $(x + 2)^2$.

Sum of Squares

The expression $x^2 + 4$ is an example of a sum of squares. It's written as the sum of two squared terms: $x^2$ and $4$. A common mistake is to think that a sum of squares can be factored similarly to a difference of squares, which is given by the formula: $a^2 - b^2 = (a - b)(a + b)$.
However, sums of squares do not factorize over the real numbers. This means you cannot break down $x^2 + 4$ into a product of two simpler binomials with real number coefficients. Instead, it remains in its original form. When dealing with sums of squares, it's crucial to remember they don't factorize unless we include complex numbers.

Expanding Binomials

To check if the original factorization is correct, let's expand the binomial $(x + 2)^2$. Using the formula for binomial expansion, $(a + b)^2 = a^2 + 2ab + b^2$, we get the following:
\[(x + 2)^2 = x^2 + 2 \times x \times 2 + 2^2 = x^2 + 4x + 4\]
By expanding, we see that the result contains an extra term of $4x$, which is not present in the original expression $x^2 + 4$. Hence, the factorization $(x + 2)^2$ is incorrect.

Real Numbers

In this context, real numbers refer to numbers that can be found on the number line. This includes all positive and negative integers, fractions, decimals, and irrational numbers. The key aspect is that sums of squares like $x^2 + 4$ do not factorize into simpler expressions involving real numbers. Some expressions require us to use imaginary numbers (like the square root of negative one, denoted as $i$) for their factorization. For example, $x^2 + 4 = (x + 2i)(x - 2i)$ involves complex numbers.
Remember that the expression $x^2 + 4$ is already in its simplest form in real number terms. Factorization mistakes often come from forgetting this key property.

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91影视

A student incorrectly factored. $$ x^{2}+4 \text { as }(x+2)^{2}$$

Short Answer

Step by step solution

Observe the Given Expression

Review the Incorrect Factorization

Expand the Incorrect Factorization

Compare the Expanded Form with the Original Expression

Conclude the Correct Factorization

Key Concepts

Incorrect Factorization

Sum of Squares

Expanding Binomials

Real Numbers

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