Math / Algebra

QuestionWhich is the correct way to factor $3 x^{2}-5 x-2$ ? $\begin{array}{l} (3 x-1)(x+2) \\ (3 x+1)(x+2) \\ (3 x-1)(x-2) \\ (3 x+1)(x-2) \end{array}$

Studdy Solution

STEP 1

1. We need to factor the quadratic polynomial $3x^2 - 5x - 2$ .
2. Factoring a quadratic polynomial of the form $ax^2 + bx + c$ involves finding two binomials $(px + q)(rx + s)$ such that their product equals the original polynomial.
3. The coefficients of the polynomial in the factored form must satisfy the conditions $a = pr$ , $b = pq + rs$ , and $c = qs$ .

STEP 2

1. Identify the correct pairs of factors for the constant term $-2$ and the leading coefficient $3$ .
2. Formulate potential factorizations based on these pairs.
3. Expand each potential factorization.
4. Compare the expanded form to the original polynomial to verify the correct factorization.

STEP 3

Identify pairs of factors for the constant term $-2$ and the leading coefficient $3$ .
For $-2$ : Possible pairs are $(1, -2)$ , $(-1, 2)$ , $(2, -1)$ , and $(-2, 1)$ . For $3$ : Possible pairs are $(1, 3)$ and $(3, 1)$ .

STEP 4

Formulate potential factorizations based on these pairs.
Potential factorizations:
1. $(3x - 1)(x + 2)$
2. $(3x + 1)(x + 2)$
3. $(3x - 1)(x - 2)$
4. $(3x + 1)(x - 2)$

STEP 5

Expand the first potential factorization to check if it matches the original polynomial.
$(3x - 1)(x + 2) = 3x \cdot x + 3x \cdot 2 - 1 \cdot x - 1 \cdot 2 = 3x^2 + 6x - x - 2 = 3x^2 + 5x - 2$

STEP 6

Expand the second potential factorization to check if it matches the original polynomial.
$(3x + 1)(x + 2) = 3x \cdot x + 3x \cdot 2 + 1 \cdot x + 1 \cdot 2 = 3x^2 + 6x + x + 2 = 3x^2 + 7x + 2$

STEP 7

Expand the third potential factorization to check if it matches the original polynomial.
$(3x - 1)(x - 2) = 3x \cdot x + 3x \cdot (-2) - 1 \cdot x - 1 \cdot (-2) = 3x^2 - 6x - x + 2 = 3x^2 - 7x + 2$

STEP 8

Expand the fourth potential factorization to check if it matches the original polynomial.
$(3x + 1)(x - 2) = 3x \cdot x + 3x \cdot (-2) + 1 \cdot x + 1 \cdot (-2) = 3x^2 - 6x + x - 2 = 3x^2 - 5x - 2$

STEP 9

Compare the expanded forms to the original polynomial $3x^2 - 5x - 2$ .
The correct factorization matches the expanded form of $(3x + 1)(x - 2)$ .
Solution: The correct factorization of $3x^2 - 5x - 2$ is $(3x + 1)(x - 2)$ .

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QuestionWhich is the correct way to factor $3 x^{2}-5 x-2$ ? $\begin{array}{l} (3 x-1)(x+2) \\ (3 x+1)(x+2) \\ (3 x-1)(x-2) \\ (3 x+1)(x-2) \end{array}$

Studdy Solution

STEP 1

STEP 2

1. Identify the correct pairs of factors for the constant term $-2$ and the leading coefficient $3$ .
2. Formulate potential factorizations based on these pairs.
3. Expand each potential factorization.
4. Compare the expanded form to the original polynomial to verify the correct factorization.

STEP 3

Identify pairs of factors for the constant term $-2$ and the leading coefficient $3$ .
For $-2$ : Possible pairs are $(1, -2)$ , $(-1, 2)$ , $(2, -1)$ , and $(-2, 1)$ . For $3$ : Possible pairs are $(1, 3)$ and $(3, 1)$ .

STEP 4

Formulate potential factorizations based on these pairs.
Potential factorizations:
1. $(3x - 1)(x + 2)$
2. $(3x + 1)(x + 2)$
3. $(3x - 1)(x - 2)$
4. $(3x + 1)(x - 2)$

STEP 5

Expand the first potential factorization to check if it matches the original polynomial.
$(3x - 1)(x + 2) = 3x \cdot x + 3x \cdot 2 - 1 \cdot x - 1 \cdot 2 = 3x^2 + 6x - x - 2 = 3x^2 + 5x - 2$

STEP 6

Expand the second potential factorization to check if it matches the original polynomial.
$(3x + 1)(x + 2) = 3x \cdot x + 3x \cdot 2 + 1 \cdot x + 1 \cdot 2 = 3x^2 + 6x + x + 2 = 3x^2 + 7x + 2$

STEP 7

Expand the third potential factorization to check if it matches the original polynomial.
$(3x - 1)(x - 2) = 3x \cdot x + 3x \cdot (-2) - 1 \cdot x - 1 \cdot (-2) = 3x^2 - 6x - x + 2 = 3x^2 - 7x + 2$

STEP 8

Expand the fourth potential factorization to check if it matches the original polynomial.
$(3x + 1)(x - 2) = 3x \cdot x + 3x \cdot (-2) + 1 \cdot x + 1 \cdot (-2) = 3x^2 - 6x + x - 2 = 3x^2 - 5x - 2$

STEP 9

Compare the expanded forms to the original polynomial $3x^2 - 5x - 2$ .
The correct factorization matches the expanded form of $(3x + 1)(x - 2)$ .
Solution: The correct factorization of $3x^2 - 5x - 2$ is $(3x + 1)(x - 2)$ .

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QuestionWhich is the correct way to factor 3x2−5x−23 x^{2}-5 x-23x2−5x−2 ? (3x−1)(x+2)(3x+1)(x+2)(3x−1)(x−2)(3x+1)(x−2)\begin{array}{l} (3 x-1)(x+2) \\ (3 x+1)(x+2) \\ (3 x-1)(x-2) \\ (3 x+1)(x-2) \end{array}(3x−1)(x+2)(3x+1)(x+2)(3x−1)(x−2)(3x+1)(x−2)​

Studdy Solution

STEP 1

STEP 2

1. Identify the correct pairs of factors for the constant term −2-2−2 and the leading coefficient 333.2. Formulate potential factorizations based on these pairs.3. Expand each potential factorization.4. Compare the expanded form to the original polynomial to verify the correct factorization.

STEP 3

Identify pairs of factors for the constant term −2-2−2 and the leading coefficient 333.For −2-2−2: Possible pairs are (1,−2)(1, -2)(1,−2), (−1,2)(-1, 2)(−1,2), (2,−1)(2, -1)(2,−1), and (−2,1)(-2, 1)(−2,1). For 333: Possible pairs are (1,3)(1, 3)(1,3) and (3,1)(3, 1)(3,1).

STEP 4

Formulate potential factorizations based on these pairs.Potential factorizations:1. (3x−1)(x+2)(3x - 1)(x + 2)(3x−1)(x+2)2. (3x+1)(x+2)(3x + 1)(x + 2)(3x+1)(x+2)3. (3x−1)(x−2)(3x - 1)(x - 2)(3x−1)(x−2)4. (3x+1)(x−2)(3x + 1)(x - 2)(3x+1)(x−2)

STEP 5

STEP 6

STEP 7

STEP 8

STEP 9

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QuestionWhich is the correct way to factor $3 x^{2}-5 x-2$ ? $\begin{array}{l} (3 x-1)(x+2) \\ (3 x+1)(x+2) \\ (3 x-1)(x-2) \\ (3 x+1)(x-2) \end{array}$

1. Identify the correct pairs of factors for the constant term $-2$ and the leading coefficient $3$ .
2. Formulate potential factorizations based on these pairs.
3. Expand each potential factorization.
4. Compare the expanded form to the original polynomial to verify the correct factorization.

Identify pairs of factors for the constant term $-2$ and the leading coefficient $3$ .
For $-2$ : Possible pairs are $(1, -2)$ , $(-1, 2)$ , $(2, -1)$ , and $(-2, 1)$ . For $3$ : Possible pairs are $(1, 3)$ and $(3, 1)$ .

Formulate potential factorizations based on these pairs.
Potential factorizations:
1. $(3x - 1)(x + 2)$
2. $(3x + 1)(x + 2)$
3. $(3x - 1)(x - 2)$
4. $(3x + 1)(x - 2)$