Solved on Jan 22, 2024

Explain the error in the student's claim that since $x^{2}-5 x-6$ has two negative terms, both factors of $c$ will be negative.

STEP 1

Assumptions
1. The student is factoring a quadratic expression of the form $ax^2 + bx + c$ .
2. The given quadratic expression is $x^2 - 5x - 6$ .
3. The student claims that both factors of $c$ (which is -6 in this case) will be negative because the expression has two negative terms.

STEP 2

Identify the error in the student's reasoning.
The student's claim that both factors of $c$ will be negative is incorrect. The factors of $c$ must multiply to give $c$ and add to give $b$ (the coefficient of $x$ ). The sign of $c$ does not necessarily determine the signs of its factors.

STEP 3

Recall the factoring process for a quadratic expression.
To factor a quadratic expression of the form $x^2 + bx + c$ , we need to find two numbers that multiply to $c$ and add up to $b$ .

STEP 4

Apply the factoring process to the given expression $x^2 - 5x - 6$ .
We need to find two numbers that multiply to $-6$ and add up to $-5$ .

STEP 5

List the pairs of factors for $-6$ .
The pairs of factors for $-6$ are: - $(1, -6)$ - $(-1, 6)$ - $(2, -3)$ - $(-2, 3)$

STEP 6

Identify the correct pair of factors that add up to $-5$ .
The pair $(1, -6)$ adds up to $-5$ .

STEP 7

Write the factors of the quadratic expression using the identified pair.
The factors of $x^2 - 5x - 6$ are $(x - 6)$ and $(x + 1)$ .

STEP 8

Explain the correct factorization.
The correct factorization of $x^2 - 5x - 6$ is $(x - 6)(x + 1)$ . This shows that one factor is negative and the other is positive, which contradicts the student's claim that both factors would be negative.

STEP 9

Conclude the error analysis.
The error the student made was assuming that the signs of the terms in the quadratic expression dictate the signs of the factors of $c$ . In fact, the signs of the factors are determined by the need to multiply to $c$ and add to $b$ . In this case, the factors of $c$ are $-6$ and $+1$ , not both negative.

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Explain the error in the student's claim that since $x^{2}-5 x-6$ has two negative terms, both factors of $c$ will be negative.

STEP 1

Assumptions
1. The student is factoring a quadratic expression of the form $ax^2 + bx + c$ .
2. The given quadratic expression is $x^2 - 5x - 6$ .
3. The student claims that both factors of $c$ (which is -6 in this case) will be negative because the expression has two negative terms.

STEP 2

Identify the error in the student's reasoning.
The student's claim that both factors of $c$ will be negative is incorrect. The factors of $c$ must multiply to give $c$ and add to give $b$ (the coefficient of $x$ ). The sign of $c$ does not necessarily determine the signs of its factors.

STEP 3

Recall the factoring process for a quadratic expression.
To factor a quadratic expression of the form $x^2 + bx + c$ , we need to find two numbers that multiply to $c$ and add up to $b$ .

STEP 4

Apply the factoring process to the given expression $x^2 - 5x - 6$ .
We need to find two numbers that multiply to $-6$ and add up to $-5$ .

STEP 5

List the pairs of factors for $-6$ .
The pairs of factors for $-6$ are: - $(1, -6)$ - $(-1, 6)$ - $(2, -3)$ - $(-2, 3)$

STEP 6

Identify the correct pair of factors that add up to $-5$ .
The pair $(1, -6)$ adds up to $-5$ .

STEP 7

Write the factors of the quadratic expression using the identified pair.
The factors of $x^2 - 5x - 6$ are $(x - 6)$ and $(x + 1)$ .

STEP 8

Explain the correct factorization.
The correct factorization of $x^2 - 5x - 6$ is $(x - 6)(x + 1)$ . This shows that one factor is negative and the other is positive, which contradicts the student's claim that both factors would be negative.

STEP 9

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Explain the error in the student's claim that since x2−5x−6x^{2}-5 x-6x2−5x−6 has two negative terms, both factors of ccc will be negative.

STEP 1

STEP 2

Identify the error in the student's reasoning.The student's claim that both factors of ccc will be negative is incorrect. The factors of ccc must multiply to give ccc and add to give bbb (the coefficient of xxx). The sign of ccc does not necessarily determine the signs of its factors.

STEP 3

Recall the factoring process for a quadratic expression.To factor a quadratic expression of the form x2+bx+cx^2 + bx + cx2+bx+c, we need to find two numbers that multiply to ccc and add up to bbb.

STEP 4

Apply the factoring process to the given expression x2−5x−6x^2 - 5x - 6x2−5x−6.We need to find two numbers that multiply to −6-6−6 and add up to −5-5−5.

STEP 5

List the pairs of factors for −6-6−6.The pairs of factors for −6-6−6 are: - (1,−6)(1, -6)(1,−6) - (−1,6)(-1, 6)(−1,6) - (2,−3)(2, -3)(2,−3) - (−2,3)(-2, 3)(−2,3)

STEP 6

Identify the correct pair of factors that add up to −5-5−5.The pair (1,−6)(1, -6)(1,−6) adds up to −5-5−5.

STEP 7

Write the factors of the quadratic expression using the identified pair.The factors of x2−5x−6x^2 - 5x - 6x2−5x−6 are (x−6)(x - 6)(x−6) and (x+1)(x + 1)(x+1).

STEP 8

Explain the correct factorization.The correct factorization of x2−5x−6x^2 - 5x - 6x2−5x−6 is (x−6)(x+1)(x - 6)(x + 1)(x−6)(x+1). This shows that one factor is negative and the other is positive, which contradicts the student's claim that both factors would be negative.

STEP 9

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Explain the error in the student's claim that since $x^{2}-5 x-6$ has two negative terms, both factors of $c$ will be negative.

Identify the error in the student's reasoning.
The student's claim that both factors of $c$ will be negative is incorrect. The factors of $c$ must multiply to give $c$ and add to give $b$ (the coefficient of $x$ ). The sign of $c$ does not necessarily determine the signs of its factors.

Recall the factoring process for a quadratic expression.
To factor a quadratic expression of the form $x^2 + bx + c$ , we need to find two numbers that multiply to $c$ and add up to $b$ .

Apply the factoring process to the given expression $x^2 - 5x - 6$ .
We need to find two numbers that multiply to $-6$ and add up to $-5$ .

List the pairs of factors for $-6$ .
The pairs of factors for $-6$ are: - $(1, -6)$ - $(-1, 6)$ - $(2, -3)$ - $(-2, 3)$

Identify the correct pair of factors that add up to $-5$ .
The pair $(1, -6)$ adds up to $-5$ .

Write the factors of the quadratic expression using the identified pair.
The factors of $x^2 - 5x - 6$ are $(x - 6)$ and $(x + 1)$ .

Explain the correct factorization.
The correct factorization of $x^2 - 5x - 6$ is $(x - 6)(x + 1)$ . This shows that one factor is negative and the other is positive, which contradicts the student's claim that both factors would be negative.